Anonymous at Fri, 24 May 2024 18:24:23 UTC No. 16191394

>>16191385
I'm just curious, do you reject the notion of continuity as a whole or just struggle with the notion of a set being uncountable?

Anonymous at Fri, 24 May 2024 18:51:53 UTC No. 16191429

>>16191394
Obviously I know what an uncountable set is; the question is can you prove to me that you know what it is and are not merely an m-zombie who acts like he knows?

Anonymous at Fri, 24 May 2024 18:55:42 UTC No. 16191437

Friendly reminder that Skolem's paradox is only a problem of first order logic. It should be trivially obvious that first order logic is insufficient as a foundation for math.

Anonymous at Fri, 24 May 2024 19:00:16 UTC No. 16191448

>>16191429
Sure, the set of all possible infinite strings of binary digits is uncountable (which in turn means the set of all numbers between 0 and 1 is uncountable, as otherwise it would be possible to have some sum of positive powers of 1/2 that isn't some number between 0 and 1).

The number of i.i.d samples of a BV random variable needed to exactly produce Brownian motion on an unbounded time interval is also uncountable (whereas it is countable for any bounded time interval).

These both basically come down to the same problem, there's no point at which whatever numerical expansion you use to define the number can have numbers which neither terminate or become periodic.

Anonymous at Fri, 24 May 2024 19:04:32 UTC No. 16191459

>>16191448
*there's no point at which every number defined by whatever numerical expansion must either terminate or repeat. Oops, had a stroke with the grammar there.

Anonymous at Fri, 24 May 2024 19:04:55 UTC No. 16191461

>>16191448
>The number of i.i.d samples of a BV random variable needed to exactly produce Brownian motion on an unbounded time interval is also uncountable
Why?

Anonymous at Fri, 24 May 2024 19:15:35 UTC No. 16191473

>>16191461
The simplest answer has to do with a concept called power spectral density, and the property of Brownian motion that its power spectral density is constant.

This means that for any bounded interval of non-zero length, you require an asymptotically growing frequency of sampling (i.e., you have to partition the bounded interval finer and finer) in order to produce Brownian motion on that interval.

Once you allow that interval to grow in an unbounded fashion, you require an uncountable number of those "slices" as there is no countable division process "fast enough" to partition the space without error. In essence, you are adding a countable number of samples that are required in an endless fashion such that the division of the "time" for the Brownian motion never actually halts.

Anonymous at Fri, 24 May 2024 19:23:26 UTC No. 16191480

>>16191473
Intredasting. I just figured that I missed the part where you demanded the random variables to be iid. Now it makes more sense. With the usual construction as a Gaussian process (not iid) of course countably many would suffice.

Anonymous at Fri, 24 May 2024 19:28:37 UTC No. 16191486

>>16191480
There's actually a somewhat neat theorem called the Stochastic Sampling Theorem which extends the Shannon-Nyquist theorem to continuous time stochastic processes.

If the samples are correlated (meaning that there is some non-zero epsilon where your covariance between x_t and x_t+epsilon will always be positive) you actually have "bandlimiting" or a compactly supported PSD automatically provided that your process is BV.

Bremaud's Fourier Analysis and Stochastic Processes has a bit about this if I'm remembering correctly.

Anonymous at Sat, 25 May 2024 01:26:26 UTC No. 16191930

>>16191385
>A mathematical zombie will defend the theorem that "the real numbers are uncountable," but what he envisions in his mind when he says "real numbers" is a countable set.
"m-zombies" don't envision sets at all, they skip the interpretation step entirely and work directly with the language, identifying [math]\mathbb{R}[/math] with the FOL expression that defines it (up to biimplication).
The ability to switch seamlessly between this formalist viewpoint and the "real" Platonic viewpoint is essential to understanding modern mathematical logic (from Lowenheim-Skolem onwards), which probably explains why so many people get filtered by the subject.

Anonymous at Sat, 25 May 2024 05:30:58 UTC No. 16192137

>>16191385
The answer is that most people think of computable numbers, which are countable.

Anonymous at Sat, 25 May 2024 05:38:17 UTC No. 16192146

>>16191385
>consider a model of ZFC in which the natural numbers have cardinality [math]\aleph_1[/math]

Anonymous at Sat, 25 May 2024 06:53:01 UTC No. 16192213

>>16191437
How do you tell the difference between someone doing higher-order logic and an m-zombie saying the same stuff but interpreting it as a many-sorted first-order logic?

Anonymous at Sat, 25 May 2024 13:15:38 UTC No. 16192565

>>16191437
no

Anonymous at Sat, 25 May 2024 13:25:20 UTC No. 16192576

>>16192146
no no, it should be one where the natural numbers have a cardinality [math]\mathfrak {c}[/math]

Anonymous at Sat, 25 May 2024 13:27:02 UTC No. 16192578

>>16192213
well, for that you'd need to not be an φ-zombie(philosophical zombie), are you sure you ain't one?

Anonymous at Sun, 26 May 2024 06:57:26 UTC No. 16193681

>>16192578
This is the Moldovan schizo that has been spamming /pol/ about AI recently, do not engage with him, he's mentally ill.

Anonymous at Sun, 26 May 2024 06:59:12 UTC No. 16193683

>>16191385
>ZFC
>ever
I'm too enlightened for that