amazingly simple set theory
Mar. 8th, 2019 01:59 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
juan_gandhiPocket 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.
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Date: 2019-03-09 02:24 am (UTC)Proof of axiom?
Axioms do not need proof, right?
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Date: 2019-03-09 02:58 am (UTC)This case is simpler. We know what "axiom of choice" is. In this theory, this statement follows from other statements.
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Date: 2019-03-09 03:13 am (UTC)What is "this" theory? "Set theory"?
> this statement
What statement?
> follows from other statements.
What "other" statements?
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Date: 2019-03-09 03:37 am (UTC)"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.
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Date: 2019-03-09 04:42 am (UTC)~~~
In "Pocket Set Theory" the formulation of "axiom of choice" follows from axioms of the theory in which we model our theory.
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In what theory do you model model "our theory"?
What is "our theory"? Is "our theory" == "Pocket Set Theory"?
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Date: 2019-03-09 05:25 am (UTC)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.
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Date: 2019-03-09 06:04 am (UTC)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"?
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Date: 2019-03-09 06:06 am (UTC)Regarding modeling the whole Zermelo-Fraenkel Set Theory in Pocket Set Theory, it's just impossible.
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Date: 2019-03-09 06:17 am (UTC)Does it mean that "Pocket Set Theory" does NOT really verify "Axiom of choice"?
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Date: 2019-03-09 06:34 am (UTC)