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In preliminary defense of the axiom, let me point out that whatever attitude toward one might harbor, nevertheless the axiom cannot be seen as incoherent or inconsistent, because Victoria Gitman and I have proved that all of my multiverse axioms are true in the multiverse consisting of the countable computably saturated models of ZFC. So the axiom is neither contradictory nor incoherent. See A natural model of the multiverse axioms.

You might be interested in the brief essay I wrote on the topic, A question for the mathematics oracle, published in the proceedings of the Singapore workshop on Infinity and Truth. For an interesting and entertaining interlude, the workshop organizers had requested that everyone at the workshop pose a specific question that might be asked of an all-knowing mathematical oracle, who would truthfully answer. My question was whether in mathematics we really do have a absolute concept of the finite.

The well-foundedness mirage axiom asserts that this phenomenon is universal: all universes are wrong about well-foundedness.

You might be interested in the brief essay I wrote on the topic, A question for the mathematics oracle, published in the proceedings of the Singapore workshop on Infinity and Truth. For an interesting and entertaining interlude, the workshop organizers had requested that everyone at the workshop pose a specific question that might be asked of an all-knowing mathematical oracle, who would truthfully answer. My question was whether in mathematics we really do have a absolute concept of the finite.

The well-foundedness mirage axiom asserts that this phenomenon is universal: all universes are wrong about well-foundedness.

In defense of the mirage axiom, let me point out that whatever attitude toward it one might harbor, nevertheless the axiom cannot be seen as incoherent or inconsistent, because Victoria Gitman and I have proved that all of my multiverse axioms are true in the multiverse consisting of the countable computably saturated models of ZFC. So the axiom is neither contradictory nor incoherent. See A natural model of the multiverse axioms.

You might be interested in the brief essay I wrote on the topic, A question for the mathematics oracle, published in the proceedings of the Singapore workshop on Infinity and Truth. For an interesting and entertaining interlude, the workshop organizers had requested that everyone at the workshop pose a specific question that might be asked of an all-knowing mathematical oracle, who would truthfully answer. My question was whether in mathematics we really do have a absolute concept of the finite.

In preliminary defense of the axiom, let me point out that whatever attitude toward one might harbor, nevertheless the axiom cannot be seen as incoherent or inconsistent, because Victoria Gitman and I have proved that all of my multiverse axioms are true in the multiverse consisting of the countable computably saturated models of ZFC. So the axiom is neither contradictory nor incoherent. See A natural model of the multiverse axioms.

You might be interested in the brief essay I wrote on the topic, A question for the mathematics oracle, published in the proceedings of the Singapore workshop on Infinity and Truth. For an interesting and entertaining interlude, the workshop organizers had requested that everyone at the workshop pose a specific question that might be asked of an all-knowing mathematical oracle, who would truthfully answer. My question was whether in mathematics we really do have a absolute concept of the finite.

Of course, within the axiomatic system of ZFC or other systems, we have a clear definition of what it means to be finite. The issue is not that, but rather the extent to which these internal accounts of finiteness agree with the naive pre-reflective accounts of the finite.

Of course, within the axiomatic system of ZFC or other systems, we have a clear definition of what it means to be finite. The issue is not that, but rather the extent to which these internal accounts of finiteness agree with the naive pre-reflective accounts of the finite as used in the meta-theory.