Date: Feb 15, 2013 2:54 PM
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
>> 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
Not widely accepted, and, I would not say original
since my eventual understanding of matters is
close to Lesniewski's.
I never have any difficulty with a proof such as
you are indicating here. That is how mathematics
is, and should be, done.
What I refer to (and should have been more precise
about) is the idea that "mathematical logic" is a
logic that has a formal deductive calculus in which
all proofs such as the one you initiated are claimed
to occur (in principle).
In that context, what does it mean to "work in ZFC"?
Perhaps mistakenly, I take that to mean one is using
the quantified statements of the theory inside of a
formal deductive calculus.
Since modern set theory, influenced by logical atomism
and minimalism, claims that there is no need for names
such as "2" in your statement above -- its signature
is either <e> or <e,=> depending on how one feels
concerning "eliminability of identity" -- one must
make the effort to understand its implementation.
In your statement above, you are designating variables
to be parameters. What I mean by this is that you are
assigning two arbitrary symbols that have no intrinsic
semantic interpretation a "definite quality".
In a formal proof, this is an application of the axiom
Now you have the epistemic problem. Of course, if
you assert distinctness
that becomes an assumption that must be discharged.
The dischargeable assumption of your statement is
much more complex.
Your use of a "constant", or "name" is something
that 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 2
Ax(x=2() <-> x=S(S(null())))
So, the truth of your deduction is not only contingent
on your dischargeable assumptions, but also on the
relationship of definite descriptions to model theory.
As I said, perhaps I am mistaken.