Re: Implementable Set Theory and Consistency of ZFC

Jesse F. Hughes wrote:

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

What happens if I say that (1-4) are the _only_ axioms, of Implementable
Set Theory. And that _only_ the sets which can be formed with those four
axioms are sets, in Implementable Set theory.

How do you plan on saying that second sentence without adding it as an
axiom? What is the logic that you are using? How do we make
inferences from the axioms?

Okay. Axiom of Completeness: (1-4) are the only axioms of Implementable
Set Theory. (Maybe I thought such Completeness was the default)

When given this additional restriction, is it true then that (5-8)
follows from (1-4), _within_ the realm of Implementable Set Theory
(my "simple model")? According to you?

Seems plausible, if only you could make clear how you plan on adding
that restriction.

As done in the article.

You seem to have an odd view of constructivism here. A constructivist
would say: To prove (Ex)(P), you must show that there is an x
satisfying P, i.e. that the axioms *prove* such an x exists.

Yes.

But you want to say: To prove ~(Ex)(P), it suffices that the axioms do
*not* prove an x satisfying P exists. At least, this is how I
interpret your claims.

http://www.cs.helsinki.fi/u/astyrman/FST.pdf ; page 3, 2nd paragraph .

Your argument holds water, it seems. I've found an analogous thought
in "Finitist Set Theory", where it is said: "The axiom of Foundation
forbids also the paradoxial sets that may have been created _without_
the axioms of FST." Should I conclude that (5-9) are _only_ derivable
for sets that were _created_ with (1-4) in IST, but not for sets that
were NOT created by employing (1-4), assuming that they nevertheless
can exist then? Okay, _then_ (5-9) might be needed as axioms, indeed,
for the latter sets, and _then_ they are not theorems. Am I close?

But this isn't any constructive principle I recognize.

Han de Bruijn

.

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