论文标题
在没有选择公理的情况下重新审视的无限组合学
Infinite Combinatorics revisited in the absence of Axiom of Choice
论文作者
论文摘要
我们研究了ZF中经典组合定理的可证明性。使用组合论点，我们为每个无限的红衣主教$κ\ in $ in $，建立以下结果 （1）$κ^+\ to（κ，ω+1）$， （2）任何家庭$ \ MATHCAL A \ subset [{on}] （3）给定一个集合映射$ f：κ\ to {[κ]}^{<ω} $，集合$κ$具有分区为$ω$ -Many $ f $ f $ fubree sets， 通过采用Karagila的绝对方法，我们证明了每个不可数的红衣主教$κ\ in $ in $， （4）给定一个集合映射$ f：κ\ to {[κ^]}^{<ω} $，有一组$ f $ - f $ - f $ f $ f $ fule （5）对于每个自然数量$ n $，每个家庭$ \ Mathcal a \ subset {[κ]}^{ω} $带有$ | a \ cap b | \ le n $ for $ \ {a，a，b \} \ in { 与（5）相反，我们表明以下语句从ZF + $ cf（ω_1）=ω_1$： （6*）每个家庭$ \ mathcal a \ subset {[ω_1]}^{ω} $带有$ | a \ cap b | \ le 1 $ 1 $ 1 $ for $ \ {a，b \} \ in {[\ MATHCAL A]}^{2} $是“本质上是”。 以下语句在ZF中不可证明，但在ZF中相当： （i）$ cf（ω_1）=ω_1$， （ii）$ω_1\ to（ω_1，ω+1）^2 $， （iii）任何家庭$ \ MATHCAL A \ subset [{on}]^{<ω} $ size $ω_1$包含一个$δ$ - 系统大小$ω_1$。 函数$ f $是“ $ω_1$上的统一世” iff $ dom（f）=ω_1$，对于每个$α<ω_1$，$ f（α）$都是从$ω$到$α$的函数。显然，存在$ω_1$的均匀降级意味着$ cf（ω_1）=ω_1$。我们证明，反向含义的失败与存在无法接近的基数相当。
We investigate the provability of classical combinatorial theorems in ZF. Using combinatorial arguments, we establish the following results for each infinite cardinal $κ\in On$, (1) $κ^+\to (κ,ω+1)$, (2) any family $\mathcal A\subset [{On}]^{<ω}$ of size $κ^+$ contains a $Δ$-system of size $κ$, (3) given a set mapping $F:κ\to {[κ]}^{<ω}$, the set $κ$ has a partition into $ω$-many $F$-free sets, By employing Karagila's method of absoluteness, we prove the following for each uncountable cardinal $κ\in On$, (4) given a set mapping $F:κ\to {[κ^]}^{<ω}$, there is an $F$-free set of cardinality $κ$, (5) for each natural number $n$, every family $\mathcal A\subset {[κ]}^{ω}$with $|A\cap B|\le n$ for $\{A,B\}\in {[\mathcal A]}^{2}$ has property $B$, In contrast to (5), we show that the following statement is not provable from ZF + $cf(ω_1)=ω_1$: (6*) every family $\mathcal A\subset {[ω_1]}^{ω}$ with $|A\cap B|\le 1$ for $\{A,B\}\in {[\mathcal A]}^{2}$ is "essentially disjoint" . The following statements are not provable in ZF, but they are equivalent in ZF: (i) $cf(ω_1)=ω_1$, (ii) $ω_1\to (ω_1,ω+1)^2$, (iii) any family $\mathcal A\subset [{On}]^{<ω}$ of size $ω_1$ contains a $Δ$-system of size $ω_1$. A function $f$ is a "uniform denumeration on $ω_1$" iff $dom(f)=ω_1$ and for every $α<ω_1$, $f(α)$ is a function from $ω$ onto $α$. It is evident that the existence of a uniform denumeration of $ω_1$ implies $cf(ω_1)=ω_1$. We prove that the failure of the reverse implication is equiconsistent with the existence of an inaccessible cardinal.
