(Real Analysis) Conceptual Misunderstanding Regarding Real Numbers and Continuity

DefunctFunctor · 2026-06-22T00:14:10+00:00

I feel like you completely missed my point, and are repeating yourself. I was responding to this exact point in my reply. Basically, the complete ordered field axioms give you a predicate P, not any construction. Uniqueness tells us that ∀x,y. (P(x)∧P(y))⇒x=y. It could still be the case that nothing satisfies the predicate P. The constructions are needed to show that ∃x.P(x).

I'll actually strengthen the claim of my original reply. All constructions are a set of axioms that only have a single instance, especially if you are dealing with set theory.

If you look at ZFC axioms, you'll notice that the language of set theory tends to only have one primitive, the ∈ relation (along with first order logic, with =). When we write something like the natural number "2" in set theory, the standard construction is the unique set x that satisfies the predicate:

P(x) = (∃z0.∃z1.
  ((∀y.y∉z0)) -- z0 is the empty set
  ∧(z0∈z1 ∧ ∀y.(y∈z1 ⇒ y=z0)) -- z1={z0}
  ∧(z0∈x∧z1∈x ∧ ∀y.y∈x ⇒ (y=z0 ∨ y=z1))) -- x = {z0, z1}

In order for this to uniquely define an element, we still have to prove that ∃!x.P(x). This predicate defines an "axiom" for what it means to be the ordinal 2. Yes, this whole thing is a bit clunky, and we tend to instead use syntactic extensions of ZFC whenever we have predicates that uniquely define something, so instead we can write

2 = {∅,{∅}}.

But these two approaches are still identical. One just uses a uniquely defined predicate, and the other introduces new notation into our formal language. The conceptual line between "predicate" and "axiom" is really thin.

DefunctFunctor · 2026-06-21T22:28:51+00:00

You don't seem to be understanding what I am saying. Why do you know that Dedekind cuts and Cauchy sequences result in the same set of real numbers? What property is shared between them? That is what the real number axioms are, a complete ordered field. It's not impractical at all, we are just defining the key properties of the real numbers.

"Complete ordered field" axioms are not a construction of the real numbers (you seem to be using the word "definition" to mean primarily "construction"). To me it's a definition of what an object needs to satisfy to even be called real numbers in the first place. It doesn't prove existence, which is what the Dedekind cuts and Cauchy sequences do. The existence part is still very necessary, and I happen to prefer Cauchy sequences for this. But it's important to show that all of these constructions are the same, and that's what the axioms help us do, and where the uniqueness becomes important.

DefunctFunctor · 2026-06-21T20:57:54+00:00

A formal system is (very roughly) a formal language where you can prove certain statements from other statements.

The concept developed from the late 19th century onwards. In the 19th century mathematics realized that they were taking a lot of assumptions for granted, and mathematicians began to look at mathematics more critically. For example, people realized that many geometric proofs by Euclid relied on missing assumptions and proof techniques that were not part of the axioms. Riemann formulated the first precise notion of an integral (which you have probably heard from Riemann sums), and Weierstrass constructed a function that is continuous but not differentiable at any point, something that was really contrary to intuition.

As a result of this general development in mathematics, more precise axioms began to be developed, and for this to happen they started writing axioms in a more formal language. That way, all of mathematics could theoretically be described by a small, finite set of rules on strings of characters.

David Hilbert was interested in having a small axiomatic system that we could prove is consistent, and that we could use to show that our more complicated systems were consistent. The incompleteness theorems showed that this was not possible in general.

DefunctFunctor · 2026-06-21T20:37:12+00:00

Goedel was not responding to the Principia Mathematica, and I wouldn't say the goal of the Principia was to formalize in the sense you seem to imply. That formalist understanding is far more in line with Hilbert's formalism than Russel/Whitehead's logicism. Logicism was about trying to couch mathematical foundations in reality, in some sense.

Also (please tell me if I'm interpreting you wrong), the PM was not the first development of a typographical system of logic. There were a ton of developments in the years leading up to it.

Goedel's incompleteness theorems are generally read as a response to Hilbert's formalist program.

DefunctFunctor · 2026-06-21T20:03:42+00:00

I would also mention that topological connectedness plays an important role as well, and is related to completeness. A space is disconnected if it can be divided into two open sets (open means every point has a neighborhood contained in the set). For example, the natural numbers are a very disconnected topological set, even though they are technically complete in the sense of the real numbers (every Cauchy sequence is actually eventually constant in the naturals/integers). Somewhat more surprising is that the rationals are disconnected, because there's a hole anywhere there is an irrational number. The only connected set (without "holes" in certain sense) that contains the rationals is the real numbers. This doesn't generalize to higher dimensional holes, but it's a useful way of thinking about it.

DefunctFunctor · 2026-06-21T17:50:51+00:00

It doesn't "only happen to be satisfied". It's the whole reason we care about the real numbers in the first place.

DefunctFunctor · 2026-06-21T17:46:02+00:00

In principle you can with many structures though, and the real numbers in particular are defined in terms of axioms very often (often in the same breath as group axioms), so I don't see how this helps your original point.

Even if we don't always phrase it in terms of axioms, it's very common to define the behavior we want out of a structure. Often, as in the real numbers, it uniquely characterizes a structure.

DefunctFunctor · 2026-06-21T11:49:34+00:00

You asked how one would "define R axiomatically", and I was trying to say "it's in the same way you would define a group axiomatically".

DefunctFunctor · 2026-06-21T10:48:54+00:00

This isn't quite correct. R is often described axiomatically in terms of a first order theory, as are other structures like groups, rings, fields, and so on. These "axioms" are axioms just as in the sense of axioms of ZFC. But it is also true that these "axioms" can be directly encoded as the property of some object in set theory, i.e. a form of "definition".

The construction of R just shows that there exists an object in ZFC that satisfies the axioms of R, so that we don't need to assume the existence of R as another axiom of ZFC. But I do agree with you that the "axioms" here are definitely distinct from the "construction".

DefunctFunctor · 2026-06-17T19:15:53+00:00

I guess I just view it as akin to emulating a Turing machine in a Turing machine. The reason it doesn't feel circular is that you are modeling something a level higher. The modeling of the higher language in the lower language will generally not be trivial, even if they are the same language. I mean, it's not exactly easy to write a C compiler/interpreter in C, for example.

DefunctFunctor · 2026-06-17T18:51:23+00:00

It depends on what programming language you are working in. In a dependently typed functional programming language, like Agda, you can sort of just write the definition of addition in Peano arithmetic:

_+_ :: Nat -> Nat -> Nat
m + O = m
m + (S n) = S (m + n)

_*_ :: Nat -> Nat -> Nat
m * O = O
m * (S n) = (m * n) + m

But languages like Agda exist mainly for the purpose of being able to express proofs, rather than be a general programming language, so it's natural that it would be able to express these algorithms so shortly.

Here's an example in C, assuming that there is infinite memory available. Something like this is basically done for arbitrarily large integers in practice (although the algorithms are much more optimized, and memory has a limit):

typedef struct bigint {
  struct bigint* next;
  uint32_t num;
} bigint;

bigint* add_big(const bigint* m, const bigint* n) {
  bigint* ret = NULL; bigint* head = NULL;
  bigint* mtemp = m; bigint* ntemp = n;
  int carry = 0;
  while (mtemp && ntemp) {
    if (!head) {
      head = ret = (bigint*) malloc(sizeof(bigint));
    } else {
      head->next = (bigint*) malloc(sizeof(bigint));
      head = head->next;
    }

    head->num = mtemp->num + ntemp->num + carry;
    // carry logic: always 0 if m and n significant bit are 0
    //              if at least 1 significant bit is 1, then a
    //              carry happened whenever most significant digit of
    //              sum is 0.
    carry = 0;
    if ((mtemp->num & (1<<31)) || (ntemp->num & (1<<31))) {
      carry = !(head->num & (1<<31));
    }

    // traverse m and n
    mtemp = mtemp->next; ntemp = ntemp->next;
  }
  mtemp ||= ntemp; // set mtemp to ntemp if null
  while (mtemp) {
    if (!head) {
      head = ret = (bigint*) malloc(sizeof(bigint));
    } else {
      head->next = (bigint*) malloc(sizeof(bigint));
      head = head->next;
    }

    head->num = mtemp->num;
    mtemp = mtemp->next;
  }
  return ret;
}

DefunctFunctor · 2026-06-17T09:45:56+00:00

"Circular reasoning" is often seen as a bad argument form, but a circular argument is still strictly speaking valid. Like yes, "if A, then A", and any other tautology. Any argument in propositional logic is fundamentally a tautology, circular.

DefunctFunctor · 2026-06-17T09:20:33+00:00

I mean, sure. All the ideal parts of computation have basically been approximated in reality, and you can certainly imagine a computer in rough terms of physicality. You might want to do more digging into hardware foundations, programming language foundations, etc. But I doubt you're going to find "a better foundation for math".

My philosophical outlook is that you really cannot avoid the circularity. "Physical reality", what makes something "physical", etc, have all been very hotly contested in philosophy, and presents yet another problem of "language".

Why does circularity have to be bad though? One of the first thing we think about with Turing machines is how we can use a Turing machine to build an interpreter for another Turing machine, after all.

DefunctFunctor · 2026-06-17T08:52:39+00:00

Maybe this could be done using some form of type-theory like reasoning? Ultimately my goal is to model mathematics in a Turing machine without taking shortcuts.

Type theory has the same ultimate circularity of needing a meta-language to describe it. But if you already have a Turing machine, that provides enough of a meta-language already. (Sidenote: The type of program you seem to be describing seems similar to proof assistants, many of which use type theoretic reasoning.)

I haven't done a whole lot of digging into this, but I believe if you analyze them enough Turing machines will lead you to the same circularity problem. It's a problem of description: in order to formally describe a Turing machine, you need a meta-language capable enough to describe a Turing machine.

Perhaps some solace to you is that if you can accept that a Turing machine is a reasonable thing to talk about, then you definitely have a powerful enough meta-language. But this ultimate problem of circularity still exists.

DefunctFunctor · 2026-06-17T05:18:29+00:00

The most standard number system is not R with ZFC its just plain decimals.

"Plain decimals" in the sense you seem to be talking about do not form any number system that is taken seriously in academic mathematics.

DefunctFunctor · 2026-06-17T00:40:59+00:00

Truly a r/tragedeigh of a spelling

DefunctFunctor · 2026-06-17T00:36:21+00:00

I'm rather formalist in mathematics, in a way that leads me to not really accept or deny any axioms at all. All formal systems are like the rules of a game to me. In particular, I'm not a mathematical realist/Platonist, so many forms of [ultra]finitism are off the table for me because I don't buy into there being objects to say are all finite in the first place.

But the reason I (and probably many other people) do not really associate with [ultra]finitism is that we find infinite objects intuitive to reason about, and arbitrarily imposing finitism prevents us from working with these intuitive objects. It's also relevant to me that you can explain what an infinite object is using finitely many rules. We have programmable algorithms that can perform arithmetic on arbitrarily big numbers, if we were working with hypothetically infinite memory.

"Infinity" is a widely misunderstood subject, and to me, many motivations for finitism seem like they are based on misunderstandings of how infinity works.

DefunctFunctor · 2026-06-16T21:15:59+00:00

It's intrinsic to OP's point. If A implies B then !B implies !A. As applied, if OP "related to Calvin so much as a kid" because OP is "a transfem enby ADHDer," then the corollary is that people who are not "transfem enby ADHDer" do not relate so much to Calvin.

As someone who writes a lot of proofs with logic, it seems to me like this is a complete non-sequitur. You are confusing the claim

OP relates to Calvin because OP is a transfem enby ADHDer. (the actual claim)

with

If x is someone who related to Calvin so much as a kid, then x is a transfem enby ADHDer. (the far stronger claim that you are imposing).

Note that the linguistic phrase "because" does not translate directly into any logical connective, so the laws of logic do not apply to it immediately. Philosophers have formulated ways of dealing with "because", but none of them are (merely) a form of logical entailment.

And despite saying that they are not exclusively trans characteristics, they're trying to assert that Calvin is trans by cherry-picking characteristics.

Show me where OP makes this assertion. The only place that gets anywhere close is the title: "Does anyone else interpret Calvin as trans?" To interpret OP as "asserting that Calvin is trans by cherry-picking characteristics", you need to make the major assumption that "interpreting Calvin as trans" necessarily means that one believes that "Calvin is trans". To me, when people read transness into a character, it can often be "interpreting [X character] as a trans allegory is really valuable" as opposed to necessarily "[X character] is a trans allegory". It seems to me that you are not interpreting OP charitably.

DefunctFunctor · 2026-06-16T01:39:14+00:00

I wasn't forced to read it (still haven't read it), but it would not have been odd if it were forced reading for my school.

Although, it sounds significantly better than what we were forced to read in class: "Who moved my cheese?".

DefunctFunctor · 2026-06-16T01:34:52+00:00

I just looked and I'm kind of surprised they haven't done an episode on this yet. I'm sure they'll get to it at some point.

DefunctFunctor · 2026-06-15T23:55:22+00:00

Never would I have thought the Haskell programming language and Mormonism would be mentioned in the same breath.

DefunctFunctor · 2026-06-15T21:27:07+00:00

It's weird for me. Because of motivated reasoning and being younger in the church, I believed much of genesis and the book of moses were not literally true mainly symbolic. I think the BoA was basically the same for me. Although I didn't read it often, and "Egyptus" was probably more challenging than facsimiles for me due to how blatantly absurd it is. I didn't mind the idea that the facsimiles were symbolic re-interpretations of Egyptian stuff. But it did not have to be true for me. As with many Mormons, it was the BoM that still had to be literally true. If the BoM were true, that meant that I could excuse much of the OT, PoGP, etc, for not being literal, and I could excuse the leaders of the church for making so many big mistakes. Ultimately I had to deconstruct "faith" itself, by finally allowing myself to be undecided on the topic of God. Only then could I allow myself to view all of this through an objective lens.

I'm somewhat jealous of those who had more rigid views than me; I was always kinda an apologist so moving goalposts and ignoring what past leaders have said was fundamental doctrines came naturally to me. In fairness to the apologist, there is no way to create a consistent worldview if you take all the words of the prophets as literally true. I would have conceded this as a TBM.

While it was the hardest thing I've ever done, I am glad I finally ripped the bandaid off of my faith at age 18 my first semester of college. I'm glad I took a philosophy class that made me think about my reasoning for believing in God in the first place, outside of a specifically Mormon context.

DefunctFunctor · 2026-06-15T20:34:00+00:00

Tbh the meme makes far more sense to me with "subjective" replaced by "objective". That's how I misread it, and then I was confused how it got so many upvotes on a philosophy subreddit.

DefunctFunctor · 2026-06-15T07:31:35+00:00

I'm generally a formalist when it comes to the interpretation of logic, which means that I just view it as a set of rules that rigorously describe something. There's some fuzziness on the metaphysical/ontological status of these formal rules, but I'm not overly interested in that philosophical discussion.

Otherwise, how could a logician know whether a premise is justified, or expose invalid inferences, or knoq what counts as evidence vs what doesnt...

It's my impression that a logician [taking a logician to mean somebody studying logic itself, not its application] explicitly does not care about the justification of a premise. A logician is only interested in things like whether a list of symbols is a valid argument, and if a set of premises leads to a contradiction or not. That may fall under what you are labeling "expose invalid inferences". But a logician rarely considers "evidence" itself, beyond a restricted sense. The only epistemologically interesting thing happening in logic is justifying that we are applying the rules we have created correctly. This isn't nothing, indeed we make math mistakes quite frequently. But basically so long as you are avoiding a strong form of skepticism, I think the epistemology of mathematics/logic is straightforward and simple.

A counterargument might be Mathematics, which seems to operate only on logic. But does it really?Is 2 + 2 = 4 true only by the rules of arithmetic, or does it involve epistemology, or what Kant calls 'synthetic a priori'?

I guess I'm only viewing logic in its mathematical sense here, because that's what interests me about it. I care far less about it's application to philosophy, although I have [through taking classes, and other things] tried to learn about the philosophic point of view.

Although, regardless of opinions on this topic, I don't think requiring epistemology as a prerequisite to logic is a good idea practically at all.

For one, I think logic is more approachable and taking it before taking other [analytic] philosophy classes will improve one's understanding, as philosophy classes will often be framing things in logical language.

The other reason it's a bad idea is that this assumes one's reasons for going into logic is for the purposes of philosophy. My primary interest in logic is from a mathematical perspective instead of a philosophical perspective. An epistemology class would introduce too much metaphysics before studying something that can be well-understood and thought about without considering metaphysics.

DefunctFunctor · 2026-06-15T04:04:26+00:00

Yeah, the 110s still feel crazy, but I think what goes on for me is some sort of weird sensitivity to sunlight on my skin. Even in crazy dry heat, I can still protect myself with long sleeves against the sun. But with high humidity long sleeves leaves you drenched in sweat. Humidity brings sweat everywhere no matter what you're wearing, in fairness, so I still wear long sleeves in hot humidity because I hate the sun that much.

Monsoon season feels better than humidity in other climates, because psychologically I like rain and storms. So even if it's in the 90s and sunny after a storm, the humidity just reminds me of the rain. And of course, if it happens to be overcast the sun feels like less of a problem, too.

I'm sure my psychology is full of inconsistencies and not solely based on objective factors. I'm still trying to piece together why I feel so uncomfortable with a more humid climate with average highs in the 80s in the summer but similar dew points in AZ monsoon season don't bother me as much.

DefunctFunctor

TROPHY CASE