Re: Implementable Set Theory and Consistency of ZFC

On 31 Okt., 14:12, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
Jesse F. Hughes wrote:
Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> writes:

So, even if I don't make use of (5-8), a proof of A from (1-4) is a
proof from (1-8) ?

Of course.

So, even if I say "there exists a Foo", then such a statement is a
valid premise for proving that the integral of 1/t from 1 to x is
ln(x) ? Weird ..

The statement can be proved in the theory consisting of the usual
axioms for real analysis and "there exists a Foo", yes. Do you think
that every theorem of ZFC uses every axiom of ZFC in its proof?

It seens we have a different picture in our mind about the meaning of an
implication A => B , as has been pinpointed by Ullrich as well. This is
what I say about in in my article:

Another philosophical note is in place, when we are saying that we "make with
an axiom" and denote this as an implication A => B. In common mathematics,
the implication => just means what is de ned by a truth table in propositional
logic. But there is another form of mathematics, called constructivism. Within
constructivist mathematics, an implication has a more "operational" meaning,
like: given A, we can construct B from A. So if we say "make with an axiom",
then it is expressed herewith that we adhere to the constructivist meaning of
an implication. End of philosophical note.

I think that you and Ullrich adhere to the common "material implication"
of mathematical logic, where there is no place for axioms that "cause" a
theorem (so to speak). In the latter sense there is no room for premises
like "there exists a Foo". The axiom of Infinity is of the latter kind.

Han de Bruijn


Let me summarize:
Assume we have a theory with certain axioms A1 and A2.
Assume we have a sentence T and a proof of T that uses
only A1 but not A2. Then your kind of logic says that
we should not say "T is true in the theory based on A1,A2"
We should only say that "T is true in the theory on A1"
You justify this because you thaink that one can dstill
a minimal set of axioms necessary for a proof and just
remove everything unnecessary.
Now assume A1 is "P and Q"
and A2 is "P and R".
Then let T be "P". T follows from A1 and T follows from A2.
But we cannot simply say that it follows from one
of the axioms specifically and that the other axiom
is definitely superfluous for any proof of T.
Moreover, your interpretation of "T can be proved from A1, A2, ...,An"
would require one to show that T cannot be proved with
any single of the axioms removed, hence you need to
present n models of A1,...,An minus one axiom such that
T is false in each of these...

hagman

.

Relevant Pages

  • Re: The Law of the Excluded Middle again (long)
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  • Re: Can ZFC prove Addition is Associative?
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