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Results tagged with axioms
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user 9550
10
votes
1
answer
1k
views
Is this set theory equivalent to ZFC?
Consider a variant of set theory with these axioms:
Extensionality,
Regularity (foundation),
Separation,
Powerset,
Axiom of Choice, and
Transitive closure of a set-like relation is set-like. …
- 6,819
13
votes
Why should we believe in the axiom of regularity?
The axioms considered so far do not exclude such sets, but such sets will never appear in the cumulative hierarchy of sets $\{V_\alpha\}_{\alpha\in ON}$, where $ON$ denotes the class of all ordinal numbers …
- 6,819
12
votes
1
answer
827
views
Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$?
Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, where … is the union of sets of axioms of all $\sf PA_\beta$, where $\beta<\alpha$
Apparently, each of $\sf PA_\alpha$ is recursively axiomatizable. …
- 6,819