amazingly simple set theory

[juan_gandhi]

Pocket Set Theory 

PST also verifies the:

• Continuum hypothesis. This follows from (5) and (6) above;
• Axiom of replacement. This is a consequence of (A4);
• Axiom of choice. Proof. The class Ord of all ordinals is well-ordered by definition. Ord and the class V of all sets are both proper classes, because of the Burali-Forti paradox and Cantor's paradox, respectively. Therefore there exists a bijection between V and Ord, which well-orders V. ∎

The well-foundedness of all sets is neither provable nor disprovable in PST.

 

Comments

[dennisgorelik]

> Axiom of choice. Proof.

Proof of axiom?
Axioms do not need proof, right?

[juan_gandhi]

When you model one theory in another, you have to prove every axiom of the theory that you model.

This case is simpler. We know what "axiom of choice" is. In this theory, this statement follows from other statements.

[dennisgorelik]

> We know what "axiom of choice" is. In this theory,

What is "this" theory? "Set theory"?

> this statement

What statement?

> follows from other statements.

What "other" statements?

[juan_gandhi]

By "this theory" I meant "Pocket Set Theory".
"This statement" - the formulation of "axiom of choice". E.g. that for a set of non-empty sets, there's a set with exactly one element in each.
"Other statements" - axioms of the theory in which we model our theory.

[dennisgorelik]

I decrypted your message to:
~~~
In "Pocket Set Theory" the formulation of "axiom of choice" follows from axioms of the theory in which we model our theory.
~~~

In what theory do you model model "our theory"?
What is "our theory"? Is "our theory" == "Pocket Set Theory"?

[juan_gandhi]

Right.

I was not specific enough. Here we don't model anything; the author just notes that what is known as AC is a theorem in this specific set theory - it follows from other axioms. What I meant was something like "take a regular set theory, model it in pocket set theory" - but that was wrong; we only talked about AC here, so there.

[dennisgorelik]

> the author just notes

Who is "the author"? You?

> "take a regular set theory, model it in pocket set theory" - but that was wrong

Do you mean it was wrong to try to model "regular set theory" in "pocket set theory"?

[juan_gandhi]

The author is Randall Holmes (see https://en.wikipedia.org/wiki/Pocket_set_theory )

Regarding modeling the whole Zermelo-Fraenkel Set Theory in Pocket Set Theory, it's just impossible.

[dennisgorelik]

> modeling the whole Zermelo-Fraenkel Set Theory in Pocket Set Theory, it's just impossible

Does it mean that "Pocket Set Theory" does NOT really verify "Axiom of choice"?

[juan_gandhi]

In ZFC the AC is an axiom. But this statement (which is an axiom in ZFC) is provable in Pcket Set Theory. Other axioms of ZFC won't work (not sure about foundation axiom, but definitely not the powerset axiom).