Re: Implementable Set Theory and Consistency of ZFC

Jesse F. Hughes wrote:

Han de Bruijn <Han.deBruijn@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

.

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