Bounding 2D Functions by Products of 1D Functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2016
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| _version_ | 1866911615425708032 |
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| author | Dorais, François Hathaway, Dan |
| author_facet | Dorais, François Hathaway, Dan |
| contents | Given sets $X,Y$ and a regular cardinal $μ$, let $Φ(X,Y,μ)$ be the statement that for any function $f : X \times Y \to μ$, there are functions $g_1 : X \to μ$ and $g_2 : Y \to μ$ such that or all $(x,y) \in X \times Y$, $$f(x,y) \le \max \{ g_1(x), g_2(y) \}.$$ In ZFC, the statement $Φ(ω_1, ω_1, ω)$ is false. However, we show the theory ZF + ``the club filter on $ω_1$ is normal'' + $Φ(ω_1, ω_1, ω)$ (which is implied by ZF + AD) implies that for every $α< ω_1$ there is a $κ\in (α,ω_1)$ such that in some inner model, $κ$ is measurable with Mitchell order $\ge α$.
There was an error in Welch's paper ``Characterizing Subsets of $ω_1$ Constructible From a Real'', which he has retracted in a personal communication. Our paper originally referenced that paper. In this version of our paper, we are not using that result. Our consistency strength upper bound has changed accordingly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1601_05454 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Bounding 2D Functions by Products of 1D Functions Dorais, François Hathaway, Dan Logic Given sets $X,Y$ and a regular cardinal $μ$, let $Φ(X,Y,μ)$ be the statement that for any function $f : X \times Y \to μ$, there are functions $g_1 : X \to μ$ and $g_2 : Y \to μ$ such that or all $(x,y) \in X \times Y$, $$f(x,y) \le \max \{ g_1(x), g_2(y) \}.$$ In ZFC, the statement $Φ(ω_1, ω_1, ω)$ is false. However, we show the theory ZF + ``the club filter on $ω_1$ is normal'' + $Φ(ω_1, ω_1, ω)$ (which is implied by ZF + AD) implies that for every $α< ω_1$ there is a $κ\in (α,ω_1)$ such that in some inner model, $κ$ is measurable with Mitchell order $\ge α$. There was an error in Welch's paper ``Characterizing Subsets of $ω_1$ Constructible From a Real'', which he has retracted in a personal communication. Our paper originally referenced that paper. In this version of our paper, we are not using that result. Our consistency strength upper bound has changed accordingly. |
| title | Bounding 2D Functions by Products of 1D Functions |
| topic | Logic |
| url | https://arxiv.org/abs/1601.05454 |