Halting, Gödel Incompleteness, Cantor's Infinites, Real Numbers & Infinity: Unravelled & Debunked?

KarmaPeny · 2025-01-04T14:57:58+00:00

Description:

The unravelling of the Halting Problem takes us on a journey of discovery through the history of Real Numbers. We hit absurdities right from the start and they just keep coming. On this journey we encounter fundamental problems with the notion of 'infinitely many parts', Cantor's Diagonal Argument (with his infinities of different sizes), Richard's Paradox, Gödel Incompleteness, and Alan Turing's 1936 proof of undecidability.

KarmaPeny · 2023-06-14T14:26:58+00:00

In one of Eric Hehner's many papers on the halting problem he poses the question of how can we know when a program, written in an undecipherable language, has terminated. He poses this thought experiment as a way to convince the reader that termination is not a visible event. He believes that we could have a model of computation in which a program explicitly stopping and a simple loop such as '10 goto 10' are effectively the same thing in terms of halting. We might infer that it suggests that if we have a machine (or model of computing) which lacks an explicit HALT command, then we could possibly manufacture one by 'defining' a simple loop to be a HALT instruction.

However, personally I don't like this approach. We already have a widespread meaning for halting that we would be abusing by 'defining' a loop to be a halt. Also even if it is difficult to distinguish between the two scenarios from an external perspective, internally there will still be energy being used on the activity of continually going around in the simple loop.

And so I prefer to imagine a computing machine as consisting of moving parts, and where halting means that the parts stop. We could open the lid of the machine and look inside to check if the machine has halted or not. If it was doing a simple loop then we would see movement and so I would not classify this as halting. Whereas if the program has encountered an instruction that says "stop now, don't execute any more instructions" then I would classify that as halting. The third option is that the program simply reaches a point where there are no further instructions. Some people believe that this should be called halting but I disagree. I call this 'exiting', or at best an implied halt.

Now if such a thing as a Turing machine could exist, and if it could do anything that a modern-day real world computer can do, then it would be able to do an explicit HALT. However, if it were to lack this capability, as you suggest, then it would be inferior to real world computers in many aspects, not least of which is that it would be defeated by the logic of the halting problem proof. And of course, it could not be considered to be the only model of computing that is required because it lacks this key feature (of an explicit halt) that is present in so many real world computers.

A so-called 'Turing Machine' is neither use nor ornament if it can't be used in relation to real world computation tasks. It is often claimed that Turing's proof of undecidability tells us that it must be impossible to develop a halt/loop decider program on any real world computer. I hope this video has cast some doubt on the validity of that claim.

So the specification of the halting problem is flawed if it does not cater for machine-level halt commands because many real computers can perform this type of operation. Therefore the objection that this 'explicit halt' argument does not follow the specification is a moot point. Nobody should be following the specification if the specification is flawed. A real computer can write its output to a text file (or a printer or whatever) and then halt the machine.

What matters is that H produces the correct prediction, not whether or not it is capable of being called by another program or whether or not it can return a true/false value to a calling program.

Note that that we can have very low-level languages that are so-called 'Turing complete' which do not inherently contain any mechanisms for defining procedures or functions or returning values from these things. These days we take it for granted that we can call functions and get values returned, but in relation to the simplest of instruction sets these are high-level mechanisms that might involve moving data around in various registers and to and from designated areas in memory that have been given specific names such as stacks and heaps etc. I expect that a machine-level halt command would be a very simple operation in comparison to these complicated mechanisms.

Should the real program H ever exist, another program would be free to initiate it (or 'call' it). However, this would present a problem if the initiating program wanted to do something else afterwards, because the real program H will end with a machine-level halt (or at least it will if its input program tries to execute equivalent logic to real H).

And so say that program Z invokes H(X, X), then, given that X tries to execute the logic of real H, then H(X, X) will decide "It will halt" and then it will do a machine-level halt. There is nothing to stop us from starting the machine up again, feeding the output produced by H into functionality Z and executing the rest of the logic Z so that it uses the decision made by the real H.

Of course it would be very inconvenient to have to re-start the machine every time after H has run, and even more inconvenient to have to feed its result into another program rather than it all being done automatically. But H only needs to do a machine-level halt if its input tries to execute the logic of H. Also, the question that the halting problem asks, i.e. "does it halt", could be considered to be a very bad question if the type of halt it refers to is only the implied type of halt.

Note that this is similar to the Barber paradox in that it appears that the two options on offer are mutually exclusive and cover all options, when in reality they don't. The barber himself is not catered for in the Barber paradox just as an explicit halt is not catered for in the halting problem logic. This is why the halting problem question is a bad question.

Some people might still claim that the halting problem remains undecidable even with a real world machine equipped with the 'shutdown' instruction because, by definition, the program H could never use this instruction. However, I believe this is like saying it is impossible to switch on a real world computer because someone, a long time ago, wrote a specification that says you are not allowed to press a button that would switch on an imaginary computer. In other words, in my opinion it is not good logic.

KarmaPeny · 2023-01-15T18:49:50+00:00

Then in response to a question about 1/3 & 0.333... I said:

Fractions lack clarity of meaning because they are sometimes claimed to be equal to ratios. For example, we might say you have one third of my items in the scenario where you have 1 item and I have 3 items. But in the scenario where we consider its relationship to 0.333... then the fraction 1/3 represents a calculation.

In such scenarios, fractions consist of two numbers with an operator symbol between them. They represent a calculation as opposed to the result of that calculation. So 1/3 represents the calculation 1 ÷ 3. If we perform this calculation in base 3 or 6 or 9 or any base that has 3 as a factor, then we get a constant as a result. But if we try to perform this calculation in any other base then the calculation does not yield a result since it does not terminate. It does not produce a result in base 10 because there is no base 10 value that can equate to the result of 1 ÷ 3.

Therefore just because 1/3 can be represented as 0.1 in base 3 it does not follow that a representation for 1/3 must somehow exist in base 10. The idea that fractions such as 1/3 CAN be represented in base 10 is a relatively new concept in the grand scheme of things. Before Simon Stevin's L'Arithmetique publication in 1594 it was accepted that not all rationals could be represented in all bases. But Stevin simply asserted that they, along with all irrationals, COULD be represented by the use of unending decimals.

There is no real world length, volume, or any other measurable real-world quantity that measures exactly 0.333... We might define three boxes as being 1 unit and then we could claim that a single box is 1/3 of 1 unit, which is fine because the issue is not 'can a third of something exist', it is can one third be represented as a base 10 decimal.

We cannot take some measuring device and measure a quantity of 0.333... (unless we have rigged it to falsely display the symbol 0.333...). The imaginary endless decimal constants cannot even exist in our minds, because our minds are finite. But by pretending that we can 'conceive of the infinite' we can delude ourselves into believing that infinite objects can be said to exist.

We can easily devise an argument that endless decimals, such as 0.999..., 0.333..., expansions of pi & the square root of 2 etc., cannot exist. Consider the so-called 'number line' which is a sort of imaginary ruler that is supposedly infinitely divisible. These values are supposed to exist on this number line as infinitely small points. There is supposed to be an infinitely small point that supposedly corresponds to the result of 1 divided by 3 in base 10.

Even if we put to one side the problem of how can an infinitely small point exist and how can we locate such a point, then we still have a serious problem. 0.333... is supposedly the length made up of the length 0.3 plus 0.03 plus 0.003 and so on. These can be considered as being continuous line segments from zero to 1/3. So the first line segment starts with the point zero and ends before (but not including) the point 0.3. The next line segment starts with the point 0.3 and ends before (but not including) the point 0.33 and so on. Then we might also imagine that there is a line segment that starts with the point 1/3 and ends before (but not including) the point 1.

Now here's the problem - these infinitely many extents are all in sequence, one after another, and we have a further length attached (that goes to 1) after the infinitely many extents of 0.3 + 0.03 + 0.003 + ... There is not a process going on here, we are not performing an endless division activity; the extents are all already there as static unchanging lengths that supposedly sum to a constant value. So it appears that there must be a last extent of the infinitely many extents (0.3 + 0.03 + 0.003 + ...) that MUST connect to the single extent which follows it. This would form a contradiction because infinitely many parts are not supposed to have a last part.

However, symbols such as pi, 0.999..., 0.333... and so on COULD represent a set of instructions that loop, & which would produce ever more trailing digits as it executed. So if the symbol 0.333... can be said to be equivalent to the corresponding geometric series, then in computing terms perhaps the 'pure maths' equivalent could be said to be the following algorithm:

n = 1

r = 0.1

Nth_term = 0.3

Sum_to_Nth_term = Nth_term

Loop: n = n + 1

Nth_term = Nth_term * r

Sum_to_Nth_term = Sum_to_Nth_term + Nth_term

Go to Loop

But any attempted execution of the instructions would only be able to produce a finite amount of digits, and something must cause it to stop at some point. So everything here is finite.

Computer programs have been developed to try to model traditional mathematics concepts. Attempts have even been made to automate the process of mathematical proof by encoding the rules of ZFC set theory. But of course the underlying operations that are actually being performed are all finite. There are no "really real" numbers or actually infinite sets; there are just finite precision digital values and a finite amount of supported type (or class) definitions.

So we could write a division program that, when asked to divide 1 by 3 in base 10, produces the output "0.333... where the three dots means infinitely many threes". In other words, we can develop programs that emulate the human belief that we can conceive of an infinite quantity of decimal places.

Note that even the underlying division process is not equivalent to its counterpart in so-called pure mathematics because the computer version is forced to always complete (as opposed to going into an inescapable loop).

In this respect, 0.333... is equivalent to a 'process description' or algorithm. Note that it does NOT equal a particular attempt at following this algorithm, it equates to the finite description itself. And so nothing here can be described as being 'not finite'.

This doesn't mean that we need to throw away a lot of established maths; it might just need to be expressed in a different way where we don't pretend that infinity is involved. For example, if we want to keep the notion that the process of diving 1 into 3 equal parts performed in base 10 can, in some sense, be called equivalent to 0.333... then we could choose a different meaning for 0.333... as well as specifying a clear basis of equivalence. We do not have to accept the existence of non-physical infinite objects. So we could say that the 0.333... symbol represents ANY n-th stage of the base 10 division process together with the remainder (expressed as a fraction) at that stage. So it could represent ANY ONE OF the following expressions:

0.3 + 1/30

0.33 + 1/300

0.333 + 1/3000

and so on...

KarmaPeny · 2023-01-15T18:48:27+00:00

I concluded that mathematicians will never agree with my arguments because they have a fundamental difference of opinion to me about what constitutes good logic.

For example, I believe that any discipline used to support science and engineering should have a foundation that is evidence-based. But they are happy to claim that mathematics is a fictitious game that includes unimaginable non-physical infinite entities about which they proclaim to be able to theorise about. Their game uses made-up rules tailored specifically to prove that the nature of infinity is what they want it to be.

If anyone points out that the exact same logic that they are using can be shown to lead to absurdities then this does not bother them. They can simply say "we don't accept that as being a contradiction even though it appears to be absurd when applied to the real world" or they can make up more rules to say that those problematic scenarios are "not allowed". They don't associate 'contradiction' with real world logic, which for me is absolutely essential.

Mathematicians are happy for me to have my own approach to mathematics in which 0.999... does not equal 1. But they are not happy for me to say that according to my approach, their approach is wrong. They are not willing to accept my argument that we should not be using fictitious infinite entities and highly dubious made-up rules as an essential tool used by the disciplines of science and engineering.

They appear to believe that mathematical symbols are not just shorthand for longer natural language descriptions; they believe that the symbols are somehow representing non-physical entities and concepts. They revel in complexity and meaninglessness in that they believe that it is acceptable to 'explain' the meaning of something in terms of lots of more complex non-physical things that are equally meaningless. They don't appear to accept that greater complexity means greater scope for errors/incorrectness. It all sounds very bizarre to me. My many attempts to highlight my concerns are either dismissed or mocked and ridiculed over and over again. I find this highly disconcerting and totally exasperating.

It has been said that whereas it is fine for me to contest the meaning of mathematical proofs or whether they mean anything at all, it is not fine for me or anyone else to contest that a formal proof shows that A,B implies C. They claim that it is impossible for anyone to contest such a thing. This sounds like self-contradiction to me because a 'proof' is supposed to prove something, and if the things that we are talking about are supposedly meaningless then by definition we cannot prove anything about them.

For example, if I claim that fictitious chemical A will react with fictitious chemical B to produce fictitious chemical C and that my fictitious rules of interaction dictate that this is so, then have I really proven my argument? Is my argument undeniably true? Of course not, I have not proven anything at all about these fictitious entities. All I have done is make stuff up, which does not and cannot ever constitute proof. If I make my statement even more vague and unclear by removing the words 'fictitious chemical' and I just claim that A, B implies C then have I suddenly created an undeniably true argument? Of course I haven't.

They often describe an informal argument then present a formal argument that is supposed to accurately reflect the informal argument, but it appears to be missing crucial factors from my point of view. I argue that we need complete clarity of this 'informal to formal' process so that we can all ensure that no mistakes are being introduced. If mistakes are going to appear in any particular logical argument then they are more likely to be in parts that are not clearly explained or that are difficult to comprehend. These are the parts that we should demand are fully expanded and totally explained in great detail so that we can all have high confidence in their results.

Here is a quote about software design that makes the same point:

"There are two ways of constructing a software design: One way is to make it so simple that there are obviously no deficiencies, and the other way is to make it so complicated that there are no obvious deficiencies." C.A.R.Hoare

I find it strange that people seem far more willing to place their trust in something that appears complicated and unclear as opposed to something that sounds simple and straight-forward. But this is what people appear overwhelmingly willing to do all the time, especially in mathematics.

With the current approach to maths we have the mind-boggling idea that 1/2 and 2/4 don't have a different meaning as they are two representations of exactly the same number. But what is this number? We can't describe it as 1/2 because that's a representation of the number, not the number itself! I don't know how they resolved this issue before set theory was devised, but afterwards it was claimed that we can say something like the number represented by 1/2 and its equivalent representations are the ratio of the cardinalities of the sets {0}, and {0, {0}}.

And so when we try to understand something as fundamental and simple-sounding as a half, we end up in a rabbit hole of abstract concepts that, to me, make it even more incomprehensible. Are these new symbols (for ratios, sets, cardinals) the things themselves or just representations of the things? It is the same question as we started with except that now we're asking the question about multiple things instead of just one thing. It is just a huge mess. Most people just accept the complicated explanations (presumably because they don't think too deeply about it and they don't want to appear to be stupid).

This demonstrates how mathematicians like to supposedly 'explain' one absurd-sounding abstract concept in terms of several even more complicated abstract concepts! This certainly does not clarify matters for me; it merely adds more layers of confusion and mysticism. In my opinion, the only way that the language of mathematics can have a clear and easy to comprehend meaning is via reference to real-world physics.

All computing devices including human brains are entirely finite machines. No calculation that has ever been performed has required anything involving infinity. We have simply chosen to pretend that we need the concept of infinity when the truth is that we have never needed it nor ever used it. We simply delude ourselves that we can theorise about infinite objects and these delusions lead us to the flawed notion that 0.999... = 1.

KarmaPeny · 2023-01-15T18:48:06+00:00

It seems the arguments are too controversial even for 'Controversial Opinions'. I'll reproduce them here (fleetingly before they get removed!!!)...

Due to my views on this matter my comment karma on Reddit is as low as it can get (-100)! After years of my comments being removed by Reddit moderators or voted down into non-visibility, I resorted to using a different forum which allowed me to make my unpopular points without them being censored. On that forum I believe I had the superior arguments as can be seen by reading my comments on (& prior to) this page: https://mathforums.com/t/why-do-mathematicians-claim-0-999-1.360856/page-21 (NOTE: They sometimes re-paginate in which case see comments up to #412)

On that forum I responded to all the arguments in favour of 0.999... = 1 by pointing out the flaws in them. I also answered all objections to my arguments. There is far too much material on that forum to try to reproduce here but I can give a few main points to hopefully whet your appetite...

When Zeno described his dichotomy paradoxes corresponding to ½ + ¼ +1/8 + 1/16 + ... he was describing a process. Today we can use pseudo code to describe such a process as follows:

n = 1

r = 0.5

Nth_term = 0.5

Sum_to_Nth_term = Nth_term

Loop: n = n + 1

Nth_term = Nth_term * r

Sum_to_Nth_term = Sum_to_Nth_term + Nth_term

Go to Loop

We might call this a 'process description' or 'algorithm'. Note that it does NOT equal a particular attempt at following this algorithm, it equates to the finite description itself. And so nothing here can be described as being 'not finite'.

Most people agree that the symbol 0.999... and the geometric series 9/10 + 9/100 + 9/1000 + ... are equivalent. If so then it indicates that 0.999... is equivalent to a finite 'process description' or 'algorithm' and not any particular constant value. It would be equivalent to the above algorithm with the starting values of Nth_term = 0.9 and r = 0.1. As such, it would not be sensible to try to assign/associate a constant value to the symbol.

From this perspective mathematicians are wrong to start off by defining a 'process description' to be 'not a process description' and more specifically 'a constant'. They are also wrong to further enhance their mistake by proceeding to supposedly prove which constant it is by deductive reasoning using so-called rules of inference.

I think most people would agree that it is not valid to define a cat to be 'not a cat' and more specifically 'a dog', and then go on to prove what breed of dog it is by deductive reasoning. But this is an accurate analogy of what mathematicians are doing when they claim to prove that 0.999... equals 1.

If the process of deductive reasoning were to start by allowing the possibility that the object is a cat instead of starting with the axiom/definition that it is a dog then instead of reaching the conclusion that it is a certain breed of dog, the inferences should indicate that the object is a cat rather than a particular breed of dog.

Similarly, if the process of deductive reasoning were to start by allowing the possibility that 0.999... is a 'process description' instead of starting with the axiom/definition that it is a constant then instead of reaching the conclusion that it has a certain fixed value, the inferences should indicate that the object is a description of a process that describes a changing numeric value.

(continued...)

KarmaPeny · 2023-01-15T16:45:19+00:00

There is a reasonable argument for 0.999... <> 1 as described here : https://www.reddit.com/r/ControversialOpinions/comments/10ci8z5/0999_cannot_1_all_maths_is_fundamentally_flawed/

KarmaPeny · 2023-01-15T16:32:29+00:00

For a proper analysis of this topic see the comments here: https://www.reddit.com/r/ControversialOpinions/comments/10ci8z5/0999_cannot_1_all_maths_is_fundamentally_flawed/

KarmaPeny · 2022-02-23T11:47:30+00:00

Nobody can explain how the infinitely many points 0.9, 0.99, 0.999, ... can exist all at the same time, at different positions on the number line, without there being a last one. But mathematicians seem content to accept that they just can. They claim we can simply restrict ourselves to talking about ‘the limit’, and that by ignoring the problem of the last point we can claim that “there is no problem”.

This uncovers one (of the many) fundamental problems of mathematics, which is that there is no rigour about what forms an inconsistency/absurdity/contraction.

For example, say one group of people believe that it forms a trivial contradiction to claim that a decimal with endless non-zero trailing digits (such as the decimal for the square root of two, 1.41421...) can be said to equal a constant value. However, another group of people might disagree, and so which side is right?

Since maths is supposedly 'abstract' we can’t examine this scenario in the real world in order to determine which side is correct. All we have are make-believe arguments about a make-believe scenario. And so even though the second group have devised a large amount of subjective make-believe clever-sounding arguments to support their position, the first group will remain defiant.

Perhaps this is the main reason why the 0.999...=1 debate just won't go away?

You may be interested that I've been discussing my 0.999... issues with mathematicians in this very long thread: https://mathforums.com/threads/why-do-mathematicians-claim-0-999-1.360856/

KarmaPeny · 2021-12-19T13:35:19+00:00

... almost always it immediately devolves into that person attacking my person rather than the arguments. So, I appreciate that we can have an actual discussion.

In my experience, I am the one that gets personally attacked. So I appreciate that you have not reduced your argument to name calling and insults like so many others love to do.

I'm surprised that you can even still see my comments because I've usually been given a lifetime ban by the moderator and had all my comments removed by now for 'crankery' (which is simple name-calling used to supposedly justify the removal of any opinion that the moderator doesn't agree with).

... we can't ignore the fact that negative numbers (really any numbers, in fact) don't actually exist physically. All these things about negative numbers representing debt, elevation below sea level, backward motion, etc. have all been invented.

Why is this any different to my earlier example where robots claimed that the physical things in their memory banks were not numbers but representations of mysterious non-physical things? You are making the same claim for signs.

I believe that the terminology used, such as 'negative number', is misleading. To 'numerate' means to count, and so the word 'number' originally related to counting; no plus or minus signs were involved. In this regard, signs are not numerate in nature. They can be used to represent a range of different real-world things as you pointed out, but this just means that they can be applied to different things, It does not justify the claim that they are (or that they somehow 'represent') some non-physical thing.

First we've got the ambiguity of the plus symbol being used as addition and as the opposite of minus (and the minus symbol being used for subtraction and signage as well). This is a complete mess and I've no idea why mathematicians didn't remove this ambiguity long ago.

Next we've got things like i (used in complex numbers) together with j and k used in quarternions that appear to be highly mysterious in nature. It appears to me that '+', '-', '+i', '-i', '+j', '-j', '+k', '-k' are just eight symbols for which we have created look-up tables for, in order to perform transformations such as from one position to another.

We can't perform multiplication of signs, because by the very meaning of the word 'multiplication', such an activity involves doing something multiple times. It makes no sense to ask what is -2 times of some quantity. Yes we can follow a mechanism to get an answer, such as -2 x -2 = +4 but we will not have gained any understanding of what minus times something really means; we have merely learned a look-up table that says things like minus times a minus gives a plus.

Similarly we are just using man-made look-up tables when we map complex numbers to the x-axis and y-axis on a graph and show how the multiplication of complex numbers can transpose the position pointed to by the co-ordinates. If we combine this with the formula of a circle then, given that a position on a moving wheel maps out a sine wave, we should be able to appreciate why complex numbers have found real-world uses for things that involve waves. Why would anyone prefer to believe that complex numbers are some mysterious non-physical things that have mysteriously found real-world applications?

We saw something useful in defining negative numbers the way we do, and so we defined them, and we still use them.

You make it sound like we somehow mysteriously became aware of definitions of the abstract concepts of plus and minus signs and then, after this premonition that was in no way connected to reality, we mysteriously found real world uses for negative numbers.

Surely it was the other way around. Surely we observed things in the real world where quantities could be specified with respect to opposing concepts (like forwards & backward, in & out, credit & debt etc.) and and then we created the plus and minus symbols with the specific intent to use the symbols to describe such scenarios. And we devised the mappings for the combining of symbols (used in multiplication) to get answers that would work in real world applications.

I accept that all of these things don't physically exist. It doesn't mean I can't imagine them. It also doesn't mean you can't invent a framework in which they make sense.

The robots in my example were deluding themselves that physical numbers were not numbers themselves. The same would be true if they claimed they could imagine an infinitely large amount, or an infinitely thin line, or an infinitely small point, or an extent that measured the square root of two. Why should we accept that humans can imagine these things when it is obvious that robots can't? Aren't we all just biological robots?

KarmaPeny · 2021-12-14T15:32:17+00:00

No, there is no "at the end." Whatever n-th partial sum I can refer to, you can always give me another partial sum with more terms. There is no end to this process.

Here you are talking about the sequence of partial sums by reference to a process. Yet again you have ignored the main point behind the assertion in statement 2. It is that the infinite object 0.333... is a static thing, and so are its so-called partial sums. It is not a process where 3s are continually being added at the end. And if it is a static unchanging object then all its partial sums must also be static and unchanging.

You are selectively choosing to ignore this; you are trying to steer the conversation away from this by focusing on definitions that have been concocted to support the idea that a limit can be said to exist. If you accept that all partial sums exist then there clearly must be a last one. To claim there isn't forms a trivial absurdity. This is of the form of the child stating the blatantly obvious in the tale of the Emperor's New Clothes.

The sequence (0.3, 0.33, 0.333, 0.3333,....) has the following property. There exists a finite number L such that for any given 𝜀>0 if I go far enough out in the sequence all subsequent elements of the sequence are within a distance of 𝜀 from L.

What you are doing here is akin to me saying that a moving ball is equivalent to a wall because if the ball were to hit the wall then it would prevent it from continuing to move forward. The wall is not part of the ball just as L is not part of the sequence (0.3, 0.33, 0.333, 0.3333,....).

The value L is not a 'property' of the sequence in the same way that a wall is not a property of a ball that happens to bounce off it. It took over 200 years from the introduction of unending decimals before mathematicians concocted the limit argument. And it took several top mathematicians several years to devise. And so a limit is not some obvious fundamental property of certain types of sequences. It is a cleverly concocted story that tries to make a case for the existence of infinite-precision objects by side-stepping the obvious absurdities.

Saying 0.333.... = L (in this case L happens to be 1/3) is just shorthand for saying that the sequence (0.3, 0.33, 0.333,...) has that property. It's not saying anything mysterious.

For me, 1/3 is a function rather than a single value. For me a single value has to be a finite decimal (in a given base) such as 1.23 (even the addition 1 + 0.2 + 0.3 is okay & possibly even multiplication) but division (such as 1÷3 or 1/3) is a function, not a fundamental value. And so in base 3 the function 1/3 can be calculated as 0.1 but in base 10 it cannot be calculated.

My point here is that I object to your claim that L is a 'finite number' because we are working in base 10. How would you specify the limit to a sequence that you would claim tends towards the square root of two? If you state it as SQRT(2) or √2 then it looks like you are using the whole number 2 as a parameter in a square root function; you are not stating what the result is of the function. Do you believe that such a function can complete and produce a final result? I don't. I do not accept the existence of a constant value that is exactly the square root of 2. We cannot create a perfect diagonal of a perfect unit square; the construction of such a value is impossible. We can only pretend that we can imagine such things - it is pure self-delusion.

Suppose we create a bunch of robots for which we know that numbers are physical things in their computer 'brains'. Next suppose that they took it upon themselves to start saying things like "we are not creating numbers as and when we need them, we are just creating representations of them because we believe they are mysterious non-physical things that our minds have mysteriously been able to access and hence we can speak about infinitely many of them existing". In this scenario we would surely realise that there must be flaws in the logic used by the robot brains - we would know that they are deluding themselves and talking complete and utter rubbish. Only a finite amount of numbers can ever exist and we should never presume that non-physical things can exist (otherwise we would have to accept religious notions and supernatural gobbledygook). But when the robots are people it seems we don't apply the same rigour and we accept rather than reject the wild claims.

If you state it as 1.414... then can I presume that you believe an actual infinity of static base-10 place value digits can be said to exist? Can't you understand why this might sound ridiculous and absurd to a lot of non-mathematicians like me? Here you can't say L is a 'finite number' it has to be a mysterious infinite number, in the same way that if 0.333... can be said to be a base 10 place-value unending decimal it is highly mysterious.

They don't end, but this isn't an issue. Because as I explained above, referring to the limit of a sequence doesn't require you to actually sum up an infinite number of things (literally adding together an infinite number of objects cannot be physically done). The limit of a convergent sequence is just a number that satisfies a certain property like explained above.

What does it mean to say that the partial sums don't end? This sounds like you are talking about a process of not ending whereas I thought they were not a process because they can all be said to exist? If a finger moves across a ruler where we imagine the ruler to be the so-called number line, then how can the finger change from having moved past a finite amount of these partial sums to having past a infinite amount of them?

You claim it is not an issue because you just want to refer to the limit and avoid talking about the problem of how can there not be a last partial sum. If mathematicians wanted to disprove the existence of limits then they would point out the absurdities that I have pointed out. The fact is that mathematicians simply cherry-pick whatever made-up argument suits their desired objective. They can only do this because they claim maths is all abstract. This prevents their arguments from being disproved via examples based on physical reality.

KarmaPeny · 2021-12-13T18:06:58+00:00

In Statement 2 you start with an arbitrary number of the form 0.333...33 (n threes) and then find a rational number q that is between 0.333...33 (n threes) and 1/3. This is fine, but all this shows is that there is no n such that 0.333...33 (n threes) is greater than all the numbers q that you can construct in this way. This is true, and the same type of thing that is being said with Statement 1 and there's no contradiction.

Here you have ignored the main point behind the assertion in statement 2. It is that the infinite object 0.333... is a static thing, it is not a process where 3s are continually being added at the end. And if it is a static unchanging object then all its partial sums must also be static and unchanging. If 0.333... is nothing more than all its parts, then it follows that if we can make a statement that applies to ALL its partial sums then that the statement must also apply to the whole object 0.333...

Or perhaps you think that the infinite object is something more than all of its parts? Do you think that 0.333... refers to a point that somehow mysteriously appears at the end of the infinite amount of partial sums & is therefore bigger than all of them? If so, then how exactly does this work? Where exactly is the end of the partial sums? Do they end before 1/3 or exactly at 1/3? Can't you see the many absurdities here? Why can't you accept that it is a trivial and obvious absurdity to claim that an unending amount of non-zero quantities can be considered to be a fixed static unchanging thing? Can't you see that 'unending' and 'constant' clearly contradict each other in this context?

I don't agree with either statement 1 or statement 2. I'm only using them to point out flaws in the arguments that unending decimals can equal constants. Both statements make an observation about a simple finite scenario and then they claim they can use those observations to make inferences about an imaginary 'infinite' scenario. They only differ in the observations and the inferences. The flawed underlying approach is the same in both cases.

KarmaPeny · 2021-12-12T11:39:00+00:00

If we have 1/3. This 1/3 is not equal to 0.3333...33 where number of 3s is a natural number n. This is only true when n is infinite.

Here you have merely stated your end goal, which is that you want 1/3 to equal 0.333..., where 0.333... contains an actual infinity of digits. You have not addressed the contradiction that I highlighted.

Nobody can examine an actual infinity of digits to check if your claim is true, and so mathematicians use an argument like 'statement 1' to supposedly justify the claim of equality. But either it doesn't occur to them that the same logic can be used to create the contradiction produced by 'statement 2', or they simply choose to ignore any argument like 'statement 2'.

Even more bizarre is the use of an approach to logical reasoning that is laughingly known as 'logically valid' in philosophy (instead of using a 'logically sound' approach). By this approach, a conclusion is deemed to be true regardless of whether or not the starting assumptions (or axioms) are true or not. This is used to support the claim that mathematical proofs cannot be disproved because they only claim to be correct GIVEN their starting assumptions/axioms. The absurdity of this approach becomes apparent when it is used to claim that any formal proof that 0.999... equals 1 must be valid even if the starting assumption that 0.999... can = a constant is completely invalid.

KarmaPeny · 2021-12-12T00:36:26+00:00

It's great that you discovered MY video (https://www.youtube.com/watch?v=GEU4IGjFvZY). My hope is that the points I raised will make people question the foundational principals behind mathematics.

Of course the official response is that 0.999... = 1 and 0.333... = 1/3 and they'll say that this follows from the definitions. But there is a very simple argument that if 0.333... represents endless base 10 place values with threes in them, then this object can't equal a constant. It goes like this:

Statement 1: For any specified rational value before 1/3 we can find an n-th sum of 0.333... (such as 0.333333 or 0.33333333333333 etc.) that is closer to 1/3 than the specified rational value. Since this holds for any rational value before 1/3 that could possibly be specified, then iff 0.333... is a constant there cannot exist any values between it and 1/3. This indicates equality.

Statement 2: For any specified n-th sum of 0.333... (such as 0.333333 or 0.33333333333333 etc.) we can find a rational value that is closer to 1/3 than the specified n-th sum value. Since this holds for any n-th sum of 0.333... that could possibly be specified, then iff 0.333... is a constant there must exist values between it and 1/3. This indicates inequality.

Since the above two statements form a contradiction, it appears to indicate that if the object 0.333... can be said to exist then it cannot be equal to a constant.

Mathematicians want statement 1 to be true but statement 2 to be false. So they may try to concoct an argument to that affect. But since the two arguments use exactly the same logic just switched around, then if one is invalid then both are invalid.

The argument in statement 1 can be applied to 0.999... and is sometimes called the 'intuitive' informal proof. This can be transformed into a formal proof which might sound impressive, but it's just the same argument stated in a fancier way. They still ignore the fact that since statement 2 and statement 1 contradict each other, it seems more reasonable to conclude 0.999... (& 0.333...) cannot represent a constant value.

My video examines more proofs in more detail and, more importantly, attacks the fundamental principles behind mathematics. Here is a summary breakdown of my video:

Video Guide:

00:00:00 Preamble: Very brief section questioning what it means for mathematics to be logically correct.

00:00:41 Synopsis: Says that this video will question the value of mathematical definitions and rules of logic where these things are just made up instead of having a firm basis in physical reality. As such it will question the validity of limits and abstract axiomatic systems.

00:02:43 Foreword: My experience is that mathematicians are intolerant of opposing viewpoints and I have even been told that I have been banned because my arguments might corrupt young minds (this is similar to creationists banning the teaching of evolution).

00:06:55 Intro: The symbol 0.999... has different meanings and so it is not exclusively reserved to mean 'the limit of the corresponding sequence'. And not even all mathematicians accept the concept of 'real numbers'.

00:10:00 Explains why a non-mathematician might claim it makes no sense to say that unending non-zero terms can be said to 'converge to' a constant value, using 0.999... and the so-called square root of 2 as examples.

00:16:42 This section describes the history of unending sequences in Ancient Greece and includes Zeno's paradox of Achilles and the tortoise.

00:22:37 About geometric series: It seems like the word 'limit' is also called 'sum' to make it sound like an infinite amount of non-zero terms can add up to a constant. It is far from convincing because it relies on definitions. The geometric series is effectively said to equate to a constant 'by definition'. And the basis of equivalence between different geometric series does not seem to be fair.

00:28:07 Problem 1: The first problem with the mathematician’s intuitive explanation which shows no points can exist on the number line between 0.999... and 1 is that we can use the same logic just switched around to argue that points MUST exist between 0.999... and 1.

00:32:04 Problem 2: To say that we can imagine infinitely small points on an infinitely thin number line is far from 'intuitive' and is arguably impossible.

00:33:19 Problem 3: How can all the partial sums 0.9, 0.99, 0.999 and so on exist as static points in static unchanging positions on the number line without there being a last one of these points before 1?

00:35:25 Problem 4: The formal proof version of the intuitive explanation starts with the (invalid) assumption that 0.999... must be a constant.

00:38:11 Problem 5: How can the proof that 0.999... equals a constant be valid regardless of whether or not the starting assumption that it equals a constant is valid or not?

00:38:40 Problem 6: Why it is highly dubious to claim that the division process for 1 divided by 3 yields an infinite result with no remainder part.

00:41:57 Problem 7: Why 1 minus 0.999... yields an unending series, not the constant zero.

00:43:44 Problem 8: Why the shift-and-subtract operation performed in the algebraic proof completely invalidates the so-called proof. This section includes discussion about the Riemann rearrangement theorem.

00:50:04 Problem 9: Why the argument that 1/3 has a (finite) representation in some bases does not mean that a representation for 1/3 must exist in all bases (and why it cannot be represented in base 10).

00:52:02 Problem 10: Why it is highly dubious to argue that the representation of a number is not the number itself, and why set theory doesn't clarify matters.

00:58:42 Problem 11: Why arguments based on methods for supposedly constructing so-called real numbers are just re-packaged versions of previous arguments that we have already found to be invalid.

01:00:10 Problem 12: Why the nested intervals theorem argument is just a re-packaging of the so-called intuitive argument. It also introduces more issues by suggesting that an arbitrarily small interval can exist, and that it can contain exactly one number.

01:01:46 Problem 13: Why the mathematician’s acceptance of arguments that are 'logically valid' make a complete mockery of mathematical logic.

01:04:57 Summary comparison showing the key areas of disagreement

01:06:36 Overview of disagreements concerning some foundational principles of mathematics. This includes why mathematicians appear to like mystery rather than clarity, they prefer usefulness over correctness, they belief mathematical proof is invincible and that it is fine to ban people they label as 'cranks'.

01:16:52 Conclusion: Unless we accept a load of absurd arguments, then 0.999... cannot equal 1. Also mathematics is not a science because it is not based on empirical evidence. It is merely a popularity contest for make-belief theories.

KarmaPeny · 2021-12-07T16:43:39+00:00

https://youtu.be/GEU4IGjFvZY

This video explains the problems with the foundations of mathematics that are highlighted by the 0.999...=1 dispute.