Searcharxiv⌕ Search

arXiv subjects

Peter Koepke

Publications and source records attributed to Peter Koepke.

8 recordsLinked to original sources
arXiv2018

An Easton-like Theorem for Zermelo-Fraenkel Set Theory with the Axiom of Dependent Choice

We show that in the theory ZF + DC + for every cardinal λ, the set of infinite subsets of λ is well-ordered (i.e., Shelah's AX4), the θ-function measuring the surjective size of the powersets P(κ) can take almost arbitrary values on any set of uncountable cardinals. This complements our results from [FK16], where we prove that in ZF (without DC), any possible behavior of the θ-function can be realized; and answers a question of Shelah in [She16], where he emphasizes that ZF + DC + AX4 is a reasonable theory, where much of set theory and combinatorics is possible.

math.LO↗
arXiv2016

An Easton-like theorem for Zermelo-Fraenkel Set Theory without Choice

We show that in Zermelo-Fraenkel Set Theory without the Axiom of Choice a surjectively modified continuum function $θ(κ)$ can take almost arbitrary values for all infinite cardinals. This choiceless version of Easton's Theorem is in sharp contrast to the situation in ZFC, where for singular cardinals $κ$, the value of $2^κ$ is strongly influenced by the behaviour of the continuum function below. Our construction can roughly be described as follows: In a ground model $V \models ZFC + GCH$ with a "reasonable" function $F: Card \rightarrow Card$ on the infinite cardinals, a class forcing ${\mathbb P}$ is introduced, which blows up the power sets of all cardinals according to $F$ . The eventual model $N \models ZF$ is a symmetric extension by ${\mathbb P}$ such that $θ^N(κ) = F(κ)$ holds for all $κ$.

math.LO↗
arXiv2013

An Easton-like Theorem for Zermelo-Fraenkel Set Theory Without Choice (Preliminary Report)

By Easton's theorem one can force the exponential function on regular cardinals to take rather arbitrary cardinal values provided monotonicity and Koenig's lemma are respected. In models without choice we employ a "surjective" version of the exponential function. We then prove a choiceless Easton's theorem: one can force the surjective exponential function on all infinite cardinals to take arbitrary cardinal values, provided monotonicity and Cantor's theorem are satisfied, irrespective of cofinalities.

math.LO↗
arXiv2007

Global Square and Mutual Stationarity at the Aleph_n

We show using a proof of the Global Square property in Core Models below a measurable of Mitchell order o(kappa)=kappa^++ (a result originally due to Jensen & Zeman) that Foreman and Magidor's Mutual Stationarity property MS(Aleph_n (1<n<omega), Cof(omega_1)) implies the existence of inner models with measurables of high Mitchell order. This MS property states that any sequence of independently chosen stationary subsets S_n of the Aleph_n (of fixed cofinality omega_1) is mutually stationary below aleph_omega.

math.LO↗
arXiv2005

Turing Computations on Ordinals

We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length $ω$ to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of Goedel's constructible universe L. This characterization can be used to prove the generalized continuum hypothesis in L.

math.LO↗
arXiv2005

The Theory of Sets of Ordinals

We propose a natural theory SO axiomatizing the class of sets of ordinals in a model of ZFC set theory. Both theories possess equal logical strength. Constructibility theory in SO corresponds to a natural recursion theory on ordinals.

math.LO↗