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2 Fixed formatting

Hello, all,

Several months ago I sat in on a seminar on AD+, which was incredibly wonderful even though I could barely follow it at all. AD+ is a technical variant of AD, the axiom of determinacy, which asserts that $$\text{For all countable } X, A\subseteq X^\omega, \text{ the game with payoff set A is determined.}$$ It is easy to show that $ZFC+AD$ is contradictory; however, assuming some large cardinals, $ZF+AD$ is consistent, and in fact has the natural model $L(\mathbb{R})$.

One of the results mentioned was the fact that a natural strengthening of AD, in which $\omega$ is replaced by an arbitrary ordinal, is inconsistent with ZF (even though assuming large cardinals, many classes of long games are determined; this I gleaned from the amazon.com preview of Itay Neeman's "Determinacy of Long Games," which I suspect has the answers to my questions, but I don't have access to it). Having lost my notes from the seminar, I have two questions:

1) What is the proof of this fact?

2) When/by whom was it proved?

Thanks to all in advance!

1

# Indeterminacy of long games

Hello, all,

Several months ago I sat in on a seminar on AD+, which was incredibly wonderful even though I could barely follow it at all. AD+ is a technical variant of AD, the axiom of determinacy, which asserts that $$\text{For all } X, A\subseteq X^\omega, \text{ the game with payoff set A is determined.}$$ It is easy to show that $ZFC+AD$ is contradictory; however, assuming some large cardinals, $ZF+AD$ is consistent, and in fact has the natural model $L(\mathbb{R})$.

One of the results mentioned was the fact that a natural strengthening of AD, in which $\omega$ is replaced by an arbitrary ordinal, is inconsistent with ZF (even though assuming large cardinals, many classes of long games are determined; this I gleaned from the amazon.com preview of Itay Neeman's "Determinacy of Long Games," which I suspect has the answers to my questions, but I don't have access to it). Having lost my notes from the seminar, I have two questions:

1) What is the proof of this fact?

2) When/by whom was it proved?

Thanks to all in advance!