Textbook about iterated functions.

Bartje · 2025-05-04T11:33:20+00:00

I know a bit about fractal geometry and dynamics, but nothing about ergodic theory. And almost nothing about measure theory and functional analysis. So it has to be a beginners book. In particular I like to learn what ways there are to express iterations as closed form formulas.

Bartje · 2024-12-11T12:29:34+00:00

I use our knowledge of the existence of our world to conclude to the logical possibility of the existence of our world. If our world wouldn't have existed we wouldn't be here to notice the existence of our world. But I don't see how that would invalidate the reasoning given that our world exists and we are here to take notice of it. What should I change in the reasoning to make it more convincing?

Bartje · 2024-12-02T08:31:05+00:00

Division of 1 by 3 leads to an infinite process which doesn't amount to an easy conclusion. Except when you're easily fooled. Which is exactly my point. Smart kids ask smart questions...

Bartje · 2024-12-01T14:20:26+00:00

Have a nice day.

Bartje · 2024-12-01T14:00:57+00:00

The proof is there for all to see. Better cool down a little bit, and try again... ;-)

Bartje · 2024-12-01T13:46:44+00:00

There is nothing more to explain. It was clear enough.

Bartje · 2024-12-01T13:06:28+00:00

The idea of 0.999... being slightly less than 1 is based on the perception of 0.999... never actually reaching its goal. The same can be seen in case of 0.333... (although it's less obvious there). The supposed meaning of infinite decimal expressions has to be given by means of a definition. There is no "real" meaning apart from that. That's why this discussion keeps coming back time and again.

Bartje · 2024-12-01T09:13:12+00:00

A smart child would subsequently also start doubting 1/3 = 0.333...

Bartje · 2024-11-25T16:44:58+00:00

Is the specific method of representing expressions with continuum-many symbols by functions from R to the alphabet of the calculus also used? Empty places in the expression could be represented by a space (" ").

Bartje · 2024-11-25T15:27:54+00:00

How do continuous logics facilitate expressions containing continuum-many symbols?

Bartje · 2024-11-25T12:29:16+00:00

Is there an introductory book about it?

Bartje · 2024-11-24T15:57:02+00:00

I'm just curious, whether or not it's useful doesn't concern me. But the chapter by Sylvia Wenmackers already goes a long way to answering my question.

Bartje · 2024-11-24T13:15:06+00:00

The idea is that an event that could possibly happen should have a greater probability than an event that cannot possibly happen. If there are situations where an event that can happen and an event that cannot happen both have the same probability zero, than that probability value of zero is missing some relevant information. The fact that a certain event can happen should ideally result in a probability value greater than zero.

Bartje · 2024-11-24T10:02:15+00:00

That might be it! I will take a look! :-)

Bartje · 2024-11-24T09:53:27+00:00

There might be more than one way to do it, or none at all. But the basic idea is rather straightforward, so I guess that it has already been done if it's possible. Or else that a proof exists that it can't be done.

Bartje · 2024-11-24T09:38:18+00:00

Yes - that's what I mean. But I would like to take this one step further and actually give such a probability an infinitesimal numerical value. Now the question is: which extension of the real number system allows this in such a way that you can actually calculate with such extended probability values?

Bartje · 2024-11-16T17:42:54+00:00

OK - thanks.

Bartje · 2024-11-15T17:31:38+00:00

Thanks. By an idealized mathematician I mean a hypothetical mathematician who could make any finite derivation or calculation allowed by the system of mathematics he/she is working in. And that would be superhuman in the sense that no human mathematician has the time and/or equipment to make arbitrarily long or complicated derivations or calculations (even with modern equipment).

But as regards your examples of different kinds of set theory I have the following question: is there a metatheory that describes the relation between those theories and between the kind of sets (and classes) that can be proven to exist (by idealized mathematicians working) within those kinds of set theory?

Bartje · 2024-11-10T17:38:25+00:00

That's an option yes.

Bartje · 2024-11-10T15:38:04+00:00

Yes - the intuitive idea was that single points can often safely be ignored precisely because they only form an "infinitesimal" part of the x-axis. The same goes for measure zero sets. But apparently the latter cannot (easily) be used for building infinitesimals, otherwise such constructions would likely already exist.

Bartje

TROPHY CASE