Re: Implementable Set Theory and Consistency of ZFC

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

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.

Sorry, I really can make no sense of this explanation.

As far as I understand constructivism in any case, if a theory T
constructively proves statement P, then so does T + X where X is any
statement. So it does not seem to me that you have addressed the
issue.

Here's what I think you *can* say: Statements (5)-(8) are true in your
model and your model is canonical in a sense (I think it's minimal).
It is not the case that (5)-(8) are therefore *entailed* by (1)-(4) in
any sense of the word that I understand.

--
"I've noticed [...] I routinely have been putting up flawed equations
with my surrogate factoring work. My take on it is that I have some
deep fear that the work is too dangerous and am sabotaging myself."
-- James S. Harris
.

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