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user21820
user21820
3:53 AM
@JadeVanadium That's kind of true.. and after all I'm not sure I believe that there really is a true model of PA in the real world (which would be infinite).. but it's interesting and highly compelling that nobody has found a problem with assuming existence of such a model, nor the fact that tons of real analysis with numerous applications are based on theorems that do seem to require ACA0, which is conservative over PA.
Completeness of FOL and JCT (Jordan curve theorem) and Brouwer's fixed-point theorem are each equivalent to WKL0 over RCA0. WKL is already a rudimentary assumption of infinity. Sequential completeness of ℝ and BW (Bolzano-Weierstrass) are each equivalent to ACA0 over RCA0. The last two are especially widely used...
Incidentally, I just noticed a statement on wikipedia that's equivalent to ACA0, which is a bit surprising to me:
> Take any countable group G and any subgroups H,I of G. Then there is a subgroup of G generated by H⋃I.
I shouldn't be surprised, because G could have arbitrary complexity. In the standard model of ACA0, G could be anywhere in the arithmetical hierarchy, and we would need the next jump to construct the group generated by H⋃I. But it's just unexpected to see it here.
 
user21820
user21820
4:26 AM
Unrelated, I'm always amazed at the lack of competency in logic on Phil SE including among those that claim to be competent. Witness this and this complete failure to understand the fixed-point lemma plus doubling down on the error. Clearly, Bumble didn't even know that the fixed-point lemma asserts the provability of an equivalence, not some equivalence...
And it's also rather disturbing in how they all copy from the internet without actually knowing any of the mathematics. For example, Double Knot, who commented on the above-linked thread (see this and this and this cheating).
 
 
3 hours later…
Jade Vanadium
Jade Vanadium
7:24 AM
@user21820 I have noticed this as well, though it seems to be a common trend in modern (amateur?) philosophy generally, not just Phil SE. After learning mathematical logic & foundations more thoroughly, I've come to feel that a lot of "philosophical" questions boil down to mere confusion about the meaning of words, and many such questions become trivial once the relevant terms are defined more precisely.
Not to say that all philosophical questions are like this, but it feels very common to me.
 
user21820
user21820
7:47 AM
@JadeVanadium Exactly! And it's not just amateur philosophy; it's indeed modern (continental) philosophy. In contrast, analytic philosophy was much better. Even Russell, who didn't understand Gödel's incompleteness theorems, eventually deferred to Gödel by telling others to listen to Gödel, but unfortunately many people today don't even bother to learn basic FOL properly, so of course they can't understand anything much.
 
Jade Vanadium
Jade Vanadium
Ah, that reminds me of an amusing tweet someone showed to me
https://twitter.com/lastpositivist/status/1775919182187512270
https://twitter.com/lastpositivist/status/1776666886018609396
Very related to the topic at hand
 
user21820
user21820
Very related!
And very incisive..
@JadeVanadium: Incidentally, I just posted an answer about precise definitions...
0
A: Is 1/3 included in the sequence 0.3, 0.33, 0.333,...?

user21820One thing that hasn't been said in the previous answers, and really needs to be emphasized, is that mathematics is not about guesswork, so we really cannot think "so it seems ok to ...". In mathematics, everything has to be precisely defined, and then whatever (precise) statement you make would b...

Also related...
0
A: Introducing function symbols in (first-order) set theory

user21820If you are working within set theory, "function" does not mean the same as "function-symbol". (Dan Christensen's suggestion is wrong.) If you work within any reasonable foundational system based on ZFC (e.g. this one), a function f ∈ ℝ→ℝ is simply a set of a certain kind that encodes the mappings...

 
 
7 hours later…
Jade Vanadium
Jade Vanadium
3:19 PM
@user21820 My understanding is that, according to our best models & observations, the universe might be intrinsically flat in such a way that would force it to be spatially infinite. In other words, there are experiments which can measure the average intrinsic curvature of the universe, and if that curvature is zero then the universe is infinite, and the measurements we've made are indistinguishable from zero curvature.
I'm really not a physicist though, so take this with a grain of salt. There is a very real possibility that I have completely misunderstood what I've read.
... To be honest though, I don't think it would really change anything even if we had experimental evidence that the physical universe we live in was both spatially and temporally finite.
Even if the space we live in is finite, it's still possible that there are disjoint spaces which are infinite. In fact, I'd say that the question of whether an infinite space is actual is not important at all, and that the mere possibility of infinity is enough. After all, a possible world is a world, it exists from the perspective of its occupants, so it exists. Modal semantics don't make our ontology any smaller; a thing which is not actual is still a thing.
There is still the question over whether infinity is possible, e.g. whether PA is consistent, or less stringently, whether the theory of successor is consistent. The weakest possible formulation of infinity (for every standard natural n, assert that there are n distinct objects) seems obviously consistent, since the formula which asserts that there are n things already has a length longer than n, and thus any proof purporting that it's false would actually be a witness that it's true.
This argument is somewhat circular for several reasons, which I suspect you'll notice immediately, but nonetheless I do think there's a very interesting observation there. The fact that the formula which asserts e.g. that there are at least 100 distinct objects requires at least 100 symbols to state, which means it's obviously impossible to disprove over FOL.
 

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