fom
Posts:
1,969
Registered:
12/4/12


Re: mathematical infinite as a matter of method
Posted:
May 4, 2013 9:59 AM


On 5/4/2013 1:21 AM, Graham Cooper wrote: > On May 4, 2:17 pm, fom <fomJ...@nyms.net> wrote: >> On 5/3/2013 8:20 PM, Graham Cooper wrote: >> >> >> >>> You don't just 'choose' a foundational theory, none of ZFC holds up >>> to Induction, when it's derived proofs are more closely examined >>> there is a lot of convoluted circular reasoning... >> >> At which point I return to the remark made earlier. The logicians >> of the late nineteenth and twentieth century went to great pains >> to reject epistemology. They are simply rediscovering it within >> their formalisms while continuing along that path of rejection. > > > Nonsense, Plato wouldn't be so unkind... > > the incompleteness theorem is just a note from mum that they couldn't > do their homework... > > ...the homework told them so... > > Q1 You can't solve this! > >> >> There are not many choices... >> >> http://en.wikipedia.org/wiki/M%C3%BCnchhausen_Trilemma >> >> At least I know why I make the choices I make. >> >> AxAy(xcy <> (Az(ycz > xcz) /\ Ez(xcz /\ ycz))) >> >> AxAy(xey <> (Az(ycz > xez) /\ Ez(xez /\ ycz))) >> >> Unfortunately, they would do you no good either. >> > > Well with ordinary subset > > xcy <> Aa aex>aey > > is sufficient to eliminate quantifiers. > > Aa ... > and > { a  ... } > > both mean " ALL A such that ... " > > When you see a 'C' > in your equations there is a hidden ALL() > ranging over all the elements
There is nothing "hidden" about it.
When a mathematician begins a proof, the statement of the premises are to be considered as true. Thus, proofs do not begin with statements containing free variables.
Meanings are hidden when one confuses the product of logical analysis  namely a language signature of undefined symbols  with the stipulation that a theory is foundational. In the latter case, the theory ought to indicate some means by which the primitive relations are recognized independent of stipulation.
There is little difference between purport and authorial intention. But, mathematical logic is supposed to frown upon intention. Rather, the goal of logical analysis is to discern an objectively recognizable underlying logical form.
> > Quantified Logic and Set Theory are equivalent theories, you don't > need the syntax of both.
When one actually uses pencil and paper, one knows what constitutes transformation rules in a proof and what constitutes statements to which those transformation rules are applied.
Get your head out of the computer.
And, for the record, your statement only holds in the sense of "set theory" obtained relative to the Russellian view. Cantor rejected the "extension of a concept" interpretation and attributed Russell's paradox to its use.
> >  > > AxAy(xcy <> (Az(ycz > xcz) /\ Ez(xcz /\ ycz))) > > > RE: ~ycz > > which just means x is PROPER subset of y > > I use not(ss(..,..)) > > in a different way, to remove an ALL() quantifier. > > ~An p(n) <> En ~p(n) > > Proof By CounterExample > >  > > e.g p(n) <> x MOD 2 = 0 > > ~An p(n) == not all numbers are even > > ......... > > PROOFBYCOUNTEREXAMPLE.PRO > > nat(0). > nat(s(X)) : nat(X). > even(0). > even(s(s(X))) : even(X). > odd(s(0)). > odd(s(s(X))) : odd(X). > > e(A,nats) : nat(A). > e(A,evens) : even(A). > e(A,odds) : odd(A). > > e(A, not(evens)) : e(A, odds). > e(A, not(odds)) : e(A, evens). > > intersects(S1,S2) : e(A,S1),e(A,S2). > not(ss( S1 , S2 )) : intersects( S1, not(S2) ). > ******************** > > > This tiny program will work out that > > ? not(ss(nats,evens)) . >> YES > > > i.e ~nats c evens > > by actually testing 0, finding it's both nat & even, > testing s(0), and finding not s(0) e evens > > Are all natural numbers even > NO! > > (this is early work since I didn't have to program ALL() yet!) > >  > > This kind of CATEGORICAL NEGATION > > if it's ODD its not EVEN > if it's BLUE it's not RED > > is used in human logic every day! > > e(A, not(evens)) : e(A, odds). > e(A, not(odds)) : e(A, evens). > >  > > PROLOG SET THEORY is the paradigm that early logicians could only > glimpse a small part of the program. >
Some, perhaps. Frege changed his mind.
"The more I have thought the matter over, the more convinced I have become that arithmetic and geometry have developed on the same basis  a geometrical one in fact  so that mathematics in its entirety is really geometry"
news://news.giganews.com:119/0NqdnRH4lKp4CFzNnZ2dnUVZ_sadnZ2d@giganews.com
Negation is eliminable.

