
Re: mathematical infinite as a matter of method
Posted:
May 4, 2013 8:30 PM


On May 4, 11:59 pm, fom <fomJ...@nyms.net> wrote: > 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_sadn...@ giganews.com > > Negation is eliminable.
You're comparing a
TACCOUNTS double journal paper accounting method lecture
to M.Y.O.B. and ORACLE ACCOUNTING PACKAGES.

What you call LOGIC doesn't work, and doesn't do anything.
Your CONCLUSIONS of X > INF
and IF HALT(S) GOTO S
are mutually supportable piffle!

X e PS(N) <> TM_N(X) halts
...
has the SAME CONSTRUCTION METHOD as Chaitans Omega
but its a COMPUTABLE POWERSET OF N.

2 birds with 1 stone!

Your EXTRA REAL is only valid with
AN INFINITE SEQUENCE OF INFERENCES.
That Paradigm is not even used with a valid set theory
(Depth First Limited)
Herc  www.BLoCKPROLOG.com

