Very large set axioms over constructive set theories
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915212197625856 |
|---|---|
| author | Jeon, Hanul Matthews, Richard |
| author_facet | Jeon, Hanul Matthews, Richard |
| contents | We investigate large set axioms defined in terms of elementary embeddings over constructive set theories, focusing on $\mathsf{IKP}$ and $\mathsf{CZF}$. Most previously studied large set axioms, notably the constructive analogues of large cardinals below $0^\sharp$, have proof-theoretic strength weaker than full Second-order Arithmetic. On the other hand, the situation is dramatically different for those defined via elementary embeddings. We show that by adding to $\mathsf{IKP}$ the basic properties of an elementary embedding $j\colon V\to M$ for $Δ_0$-formulas, which we will denote by $\mathsf{Δ_0\text{-}BTEE}_M$, we obtain the consistency of $\mathsf{ZFC}$ and more. We will also see that the consistency strength of a Reinhardt set exceeds that of $\mathsf{ZF+WA}$. Furthermore, we will define super Reinhardt sets and $\mathsf{TR}$, which is a constructive analogue of $V$ being totally Reinhardt, and prove that their proof-theoretic strength exceeds that of $\mathsf{ZF}$ with choiceless large cardinals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_05831 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Very large set axioms over constructive set theories Jeon, Hanul Matthews, Richard Logic Primary 03E70, Secondary 03E55 We investigate large set axioms defined in terms of elementary embeddings over constructive set theories, focusing on $\mathsf{IKP}$ and $\mathsf{CZF}$. Most previously studied large set axioms, notably the constructive analogues of large cardinals below $0^\sharp$, have proof-theoretic strength weaker than full Second-order Arithmetic. On the other hand, the situation is dramatically different for those defined via elementary embeddings. We show that by adding to $\mathsf{IKP}$ the basic properties of an elementary embedding $j\colon V\to M$ for $Δ_0$-formulas, which we will denote by $\mathsf{Δ_0\text{-}BTEE}_M$, we obtain the consistency of $\mathsf{ZFC}$ and more. We will also see that the consistency strength of a Reinhardt set exceeds that of $\mathsf{ZF+WA}$. Furthermore, we will define super Reinhardt sets and $\mathsf{TR}$, which is a constructive analogue of $V$ being totally Reinhardt, and prove that their proof-theoretic strength exceeds that of $\mathsf{ZF}$ with choiceless large cardinals. |
| title | Very large set axioms over constructive set theories |
| topic | Logic Primary 03E70, Secondary 03E55 |
| url | https://arxiv.org/abs/2204.05831 |