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 26 Jun 2010 15:51 On Jun 25, 5:14 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > Who has proved PA consistent using ZFC? If it were possible then I > > assume someone would have done it. It certainly would be a very > > educational exercise. > > So why not have a try at it? You'll find all the details you need in any > decent text. Name one - including page numbers - that includes that proof. Just one. C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Charlie-Boo on 26 Jun 2010 20:33 On Jun 14, 10:41 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 14, 11:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > ZFC/PA > > You're AN IDIOT. > > ZFC IS ONE thing. > PA IS ANOTHER, SIMPLER thing. > ZFC is A SET theory. > PA is a theory OF ARITHMETIC. > > First-order theories in general cannot prove THEMSELVES consistent, > so first-order PA cannot prove that first-order PA is consistent. > First-order ZFC, however, IS A STRONGER THEORY (it has an > axiom of infinity), SO IT CAN AND DOES prove that PA is consistent. > > Your problem is that you presumed to talk about ZFC/PA like it was one > thing. > Your problem is that you flaunted the fact that YOU ARE IGNORANT OF > THE > RELEVANT DIFFERENCES BETWEEN THE TWO. > > ZFC is a set theory. PA is a number theory. > PA doesn't know what an infinite set is. ZFC does. > That is the main reason why ZFC can prove that PA is consistent > (a model of PA *has* to be infinite, and PA can't prove that anything > is infinite, since in its standard model, NOTHING IS). Ok, how does ZFC prove PA consistent? I would think that the set axioms of ZFC would not help us with something that has nothing to do with what a set is in general. That's the only difference between the two. C-B
From: Charlie-Boo on 26 Jun 2010 20:44 On Jun 14, 10:41 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 14, 11:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > ZFC/PA > > You're AN IDIOT. > > ZFC IS ONE thing. > PA IS ANOTHER, SIMPLER thing. > ZFC is A SET theory. > PA is a theory OF ARITHMETIC. > > First-order theories in general cannot prove THEMSELVES consistent, > so first-order PA cannot prove that first-order PA is consistent. > First-order ZFC, however, IS A STRONGER THEORY (it has an > axiom of infinity), SO IT CAN AND DOES prove that PA is consistent. > > Your problem is that you presumed to talk about ZFC/PA like it was one > thing. > Your problem is that you flaunted the fact that YOU ARE IGNORANT OF > THE > RELEVANT DIFFERENCES BETWEEN THE TWO. > > ZFC is a set theory. PA is a number theory. > PA doesn't know what an infinite set is. ZFC does. > That is the main reason why ZFC can prove that PA is consistent > (a model of PA *has* to be infinite, and PA can't prove that anything > is infinite, since in its standard model, NOTHING IS). Prove that something is infinite with a program, you ask. Lemme see... The statement "P() is infinite." is (aA)(eB)LT(A,B)^P(B). We can prove that (aA)(eB)B=suc(A). So if P(B) is LT(A,B) we have (aA) (eB)LT(A,B) which is implied. Then the set is the set of numbers that have a successor. (eA)suc(x)=A And we have (aA)(eB)suc(A)=B So if P(a) is (eA)suc(a)=A then "P() is infinite." is (aA)(eB)LT(A,B)^ suc(A)=B and since suc=>LT we have (aA)(eA)LT(A,B) and since (aA) (eB)sub(A)=B we have (aA)P(A) and so P() is infinite. (Something like that, I'm sure.) Y~? C-B
From: Charlie-Boo on 26 Jun 2010 20:56 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; I didn't say that it is also a claim > that I make. So you don't claim it but you do claim that many people claim it. Do you claim that many people claim that they claim it? Or many that claim that they don't claim it (e.g. you)? And do you claim that there are many people who claim that you claim that they claim that you claim it? See my "1,000 Paradoxes" post where I list 1,000 English sentences that show what a real formalization of the Liar contains. "This is true of 'It is not true of this.'" etc. C-B > 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 26 Jun 2010 21:04
On Jun 24, 5:38 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > 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 So you would agree that "I always tell the truth."? (Ambiguity...) So you might say "I always tell the truth."? C-B "I made a misstatement and I stand by all my misstatements." - Dan Quayle and I don't obligate myself to > defend your wording. > > MoeBlee- Hide quoted text - > > - Show quoted text - |