How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Charlie-Boo on 24 Jun 2010 14:01 What does ZFC/PA really provide (mathematically and psychologically)? PA is a programming language - an enumeration of the r.e. sets (recursive objects.) And the ZFC part is the DATA STRUCTURES of this "programming language". When programmers need to go beyond aleph-1 integers they use local arrays or built-in structures like representations of a subset of the real numbers. We need an infinite number of variable names. So we call it an array using one name and an infinite number of subscripts. (It's really unbounded as far as we know but finite.) So what does a programming language with data structures provide you? You can run a program and maybe see that it halts. And if it halts, we have a proof that it halts. And when a program halts, what does that prove? There are a lot of things it could prove (e.g. Fermat and Goldbach) so some people say it can do most anything we can do. But the Theory of Computation tells you there are some questions that even a solution (oracle) to the Halting Problem can't provide. But I guess they would just claim that this isn't "ordinary mathematics". *sigh* C-B On Jun 14, 11:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > ZFC/PA is supposed to do all ordinary mathematics*. But it is easy to > prove that PA is consistent (its axioms and rules preserve truth) yet > (by Godel-2) PA can't do such a simple proof as that. > > Since PA can't prove something as simple as that, how could anyone be > so stupid as to claim ZFC/PA is a good basis for all of our ordinary > math? > > C-B > > Answer: I failed to "Never underestimate the stupidity of academia." > > * Putting aside the fact that this ill-defined, ridiculous association > of the temporal with the timeless is a pitiful attempt by the Prof.s > of the world to yet again say they have captured all of Math despite > Godel. (Hilbert Lives!)
From: MoeBlee on 24 Jun 2010 16:19 On Jun 24, 12:47 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > On Jun 14, 11:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > Charlie-Boo <shymath...(a)gmail.com> writes: > > > Since PA can't prove something as simple as that, how could anyone be > > > so stupid as to claim ZFC/PA is a good basis for all of our ordinary > > > math? > > > Who makes this claim? > > MoeBlee What I've said on the subject is in my own words and is not properly simplified to "ZFC/PA is a good basis for all of our ordinary math", especially as I don't know what is supposed to be indicated by 'ZFC/ PA' in such slash notation. (PA is embeded in ZFC, of course.) For example, recently I said, "The common claim is that ZFC axiomatizes all (or virtually all) ordinary mathematics. " But I did not say that I personally make that common claim. I merely said what the common claim IS; I didn't say that it is also a claim that I make. And I didn't say anything about ZFC being a "good" basis. Good in what sense? ZFC has certain merits and (arguably) certain drawbacks. It may be a suitable theory in certain ways, but I did not claim that it is simply "good" as a basis. Also, I allowed that a reasonable view of the common claim may include that only VIRTUALLY all of ordinary mathematics may be axiomatized by ZFC. MoeBlee
From: Charlie-Boo on 24 Jun 2010 16:58 On Jun 24, 4:19 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 24, 12:47 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > On Jun 14, 11:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > Charlie-Boo <shymath...(a)gmail.com> writes: > > > > Since PA can't prove something as simple as that, how could anyone be > > > > so stupid as to claim ZFC/PA is a good basis for all of our ordinary > > > > math? > > > > Who makes this claim? > > > MoeBlee > > What I've said on the subject is in my own words and is not properly > simplified to "ZFC/PA is a good basis for all of our ordinary math", > especially as I don't know what is supposed to be indicated by 'ZFC/ > PA' in such slash notation. (PA is embeded in ZFC, of course.) > > For example, recently I said, "The common claim is that ZFC > axiomatizes all (or virtually all) ordinary mathematics. " > But I did not say that I personally make that common claim. I merely > said what the common claim IS; And that being a common claim is what I claimed, so you claimed the same thing that I claimed. C-B (your Claim-Buddy) > I didn't say that it is also a claim > that I make. > > And I didn't say anything about ZFC being a "good" basis. Good in what > sense? ZFC has certain merits and (arguably) certain drawbacks. It may > be a suitable theory in certain ways, but I did not claim that it is > simply "good" as a basis. > > Also, I allowed that a reasonable view of the common claim may include > that only VIRTUALLY all of ordinary mathematics may be axiomatized by > ZFC. > > MoeBlee
From: Charlie-Boo on 24 Jun 2010 17:01 On Jun 14, 11:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > Since PA can't prove something as simple as that, how could anyone be > > so stupid as to claim ZFC/PA is a good basis for all of our ordinary > > math? > Who makes this claim? You're hallucinating. Read the first 3 sentences of Godel's famous 1931 article (not famous enough, unfortunately.) While you're at it, maybe even read more than 3 sentences. C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 24 Jun 2010 17:38
On Jun 24, 3:58 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > On Jun 24, 4:19 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > > > On Jun 24, 12:47 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > > On Jun 14, 11:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > > Charlie-Boo <shymath...(a)gmail.com> writes: > > > > > Since PA can't prove something as simple as that, how could anyone be > > > > > so stupid as to claim ZFC/PA is a good basis for all of our ordinary > > > > > math? > > > > > Who makes this claim? > > > > MoeBlee > > > What I've said on the subject is in my own words and is not properly > > simplified to "ZFC/PA is a good basis for all of our ordinary math", > > especially as I don't know what is supposed to be indicated by 'ZFC/ > > PA' in such slash notation. (PA is embeded in ZFC, of course.) > > > For example, recently I said, "The common claim is that ZFC > > axiomatizes all (or virtually all) ordinary mathematics. " > > > But I did not say that I personally make that common claim. I > merely > > said what the common claim IS; > > And that being a common claim is what I claimed, so you claimed the > same thing that I claimed. You claim that they are the same claim, though I noted specific differences. I don't need to argue whether they are the same, but only I note that I stand by my own wording and I don't obligate myself to defend your wording. MoeBlee |