```Date: Feb 15, 2013 2:54 PM
Author: fom
Subject: Re: infinity can't exist

On 2/15/2013 8:56 AM, Jesse F. Hughes wrote:> fom <fomJUNK@nyms.net> writes:>>> Here is where my complexity when interpreting>> Aristotle's remarks arise:>>>> Something like>>>> x=y>>>> is called informative identity.>>>> But, within a formal proof, one can never>> write that as an assumption.  In a formal>> proof, one begins with true sentences and>> ends with true sentences.  So all the>> metalogical analysis of meaning outside>> of a deduction is "armchair quarterbacking">> whereas what happens in a proof is>> "regulation time".>> I'm afraid I have no idea what you mean here.  Natural deduction> proofs involve assumptions all the darned time.  For instance, one may> begin the proof that sqrt(2) is irrational by:>>     Assume that a and b are natural numbers such that (a/b)^2 = 2.>> Now, given what you say below (snipped, since I don't really> understand what you're getting at and have no comment), it may be that> you think such a statement involves quantifiers and so is an exception> to the rule you state above.  But, honestly, there is *NOTHING* in the> usual presentation of natural deduction that entails an equation> cannot be taken as an assumption.>> Do you think that these comments of yours are widely accepted or> original?>>Not widely accepted, and, I would not say originalsince my eventual understanding of matters isclose to Lesniewski's.I never have any difficulty with a proof such asyou are indicating here.  That is how mathematicsis, and should be, done.What I refer to (and should have been more preciseabout) is the idea that "mathematical logic" is alogic that has a formal deductive calculus in whichall proofs such as the one you initiated are claimedto occur (in principle).In that context, what does it mean to "work in ZFC"?Perhaps mistakenly, I take that to mean one is usingthe quantified statements of the theory inside of aformal deductive calculus.Since modern set theory, influenced by logical atomismand minimalism, claims that there is no need for namessuch as "2" in your statement above --  its signatureis either <e> or <e,=> depending on how one feelsconcerning "eliminability of identity" -- one mustmake the effort to understand its implementation.In your statement above, you are designating variablesto be parameters.  What I mean by this is that you areassigning two arbitrary symbols that have no intrinsicsemantic interpretation a "definite quality".In a formal proof, this is an application of the axiomof reflexivenessa=ab=bNow you have the epistemic problem.  Of course, ifyou assert distinctnessa=ab=b|-(a=b)that becomes an assumption that must be discharged.The dischargeable assumption of your statement ismuch more complex.Your use of a "constant", or "name" is somethingthat requires introduction through definite descriptions,Definition of bottom:Ax(x=null() <-> Ay(-(xcy <-> x=y)))Assumption of bottom:ExAy(-(xcy <-> x=y))Definition of successor function:AxAy(x=S(y) <-> (Ez(ycz) /\ Az(zex <-> (zey \/ z=y))))Assumption of successor set:Ax(Ey(xcy) -> Ey(Az(zey <-> (zex \/ z=x)) /\ Ez(ycz)))Definition of 2Ax(x=2() <-> x=S(S(null())))So, the truth of your deduction is not only contingenton your dischargeable assumptions, but also on therelationship of definite descriptions to model theory.As I said, perhaps I am mistaken.
```