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amazingly simple set theory
Mar. 8th, 2019 01:59 pmPocket 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.
Structural Set Theory: objects, functions, relations
Material Set Theory: sets consist of elements, collecting them without the elements' knowledge
Material Set Theory: sets consist of elements, collecting them without the elements' knowledge
Like, here, on mathoverflow .
Leinster wrote in 2012: "rethinking".
People went crazy.
And it seems like nobody fucking knows there's a difference between a theory and a model.
That's how Chaitin was stating that there are "mathematical truths" that are true but don't follow from axioms.
For god's sake, it's been 90 years since Gödel explained it all. And? Anybody asked, how come ZFC is not based on ZFC? (sigh)
Leinster wrote in 2012: "rethinking".
People went crazy.
And it seems like nobody fucking knows there's a difference between a theory and a model.
That's how Chaitin was stating that there are "mathematical truths" that are true but don't follow from axioms.
For god's sake, it's been 90 years since Gödel explained it all. And? Anybody asked, how come ZFC is not based on ZFC? (sigh)
