What is the theory ZFC without power set?
We show that the theory ZFC, consisting of the usual axioms of ZFC but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered—is weaker than commonly supposed and...
| Main Authors: | , , |
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| Formato: | Journal article |
| Idioma: | English |
| Publicado: |
Wiley
2016
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| _version_ | 1826306551734861824 |
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| author | Gitman, V Hamkins, J Johnstone, T |
| author_facet | Gitman, V Hamkins, J Johnstone, T |
| author_sort | Gitman, V |
| collection | OXFORD |
| description | We show that the theory ZFC, consisting of the usual axioms of ZFC but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of ZFC in which ω 1 is singular, in which every set of reals is countable, yet ω 1 exists, in which there are sets of reals of every size ℵ n , but none of size ℵ ω , and therefore, in which the collection axiom fails; there are models of ZFC for which the Łoś theorem fails, even when the ultrapower is well-founded and the measure exists inside the model; there are models of ZFC for which the Gaifman theorem fails, in that there is an embedding j : M → N of ZFC models that is Σ 1 -elementary and cofinal, but not elementary; there are elementary embeddings j : M → N of ZFC- models whose cofinal restriction j : M → υ j”M is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC to a Σ 1 -formula or a Π 1 -formula is not closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory ZFC - , obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach. |
| first_indexed | 2024-03-07T06:49:39Z |
| format | Journal article |
| id | oxford-uuid:fc1a16c9-b18a-44f3-a568-55e8c78608ed |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:49:39Z |
| publishDate | 2016 |
| publisher | Wiley |
| record_format | dspace |
| spelling | oxford-uuid:fc1a16c9-b18a-44f3-a568-55e8c78608ed2022-03-27T13:18:22ZWhat is the theory ZFC without power set?Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:fc1a16c9-b18a-44f3-a568-55e8c78608edEnglishSymplectic Elements at OxfordWiley2016Gitman, VHamkins, JJohnstone, TWe show that the theory ZFC, consisting of the usual axioms of ZFC but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of ZFC in which ω 1 is singular, in which every set of reals is countable, yet ω 1 exists, in which there are sets of reals of every size ℵ n , but none of size ℵ ω , and therefore, in which the collection axiom fails; there are models of ZFC for which the Łoś theorem fails, even when the ultrapower is well-founded and the measure exists inside the model; there are models of ZFC for which the Gaifman theorem fails, in that there is an embedding j : M → N of ZFC models that is Σ 1 -elementary and cofinal, but not elementary; there are elementary embeddings j : M → N of ZFC- models whose cofinal restriction j : M → υ j”M is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC to a Σ 1 -formula or a Π 1 -formula is not closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory ZFC - , obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach. |
| spellingShingle | Gitman, V Hamkins, J Johnstone, T What is the theory ZFC without power set? |
| title | What is the theory ZFC without power set? |
| title_full | What is the theory ZFC without power set? |
| title_fullStr | What is the theory ZFC without power set? |
| title_full_unstemmed | What is the theory ZFC without power set? |
| title_short | What is the theory ZFC without power set? |
| title_sort | what is the theory zfc without power set |
| work_keys_str_mv | AT gitmanv whatisthetheoryzfcwithoutpowerset AT hamkinsj whatisthetheoryzfcwithoutpowerset AT johnstonet whatisthetheoryzfcwithoutpowerset |