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Ali Enayat and I are currently preparing a paper whose main theorem is that with respect to definable classes, Ord is NOT weakly compact. In particular, we show, in every model of ZFC,

• there is a definable Ord-tree with no definable cofinal branch.
• there is a definable 2-coloring of $V$, with no definable homogeneous proper class.
• there is a definable set-satisfiable $L_{\text{Ord},\omega}$-theory, which has no definable class model.

This result surprised me very much, since it shows that with respect to definable classes, we can prove that Ord fails to have a large cardinal property that reasonable people might have expected to hold true.

The preprint is not ready yet, but I expect that it will be ready in a few weeks. I'll post a link later, when it is available.

Ali Enayat and I have proved that with respect to definable classes, Ord is NOT weakly compact. In particular, we show, in every model of ZFC,

• there is a definable Ord-tree with no definable cofinal branch.
• there is a definable 2-coloring of a definable proper class, with no definable homogeneous proper class.
• there is a definable set-satisfiable $L_{\text{Ord},\omega}$-theory, which has no definable class model.

This result surprised me very much, since it shows that with respect to definable classes, we can prove that Ord fails to have a large cardinal property that reasonable people might have expected to hold true.

The article is now available:

• A. Enayat and J. D. Hamkins, ZFC proves that the class of ordinals is not weakly compact for definable classes, manuscript under review. (arχiv, blog post)

Ali Enayat and I are currently preparing a paper whose main theorem is that with respect to definable classes, Ord is NOT weakly compact. In particular, we show, in every model of ZFC,

• there is a definable Ord-tree with no definable cofinal branch.
• there is a definable 2-coloring of $V$, with no definable homogeneous proper class.
• there is a definable set-satisfiable $L_{\text{Ord},\omega}$-theory, which has no definable class model.

This result surprised me very much, since it shows that with respect to definable classes, we can prove that Ord fails to have a large cardinal that reasonable people might have expected to hold true.

The preprint is not ready yet, but I expect that it will be ready in a few weeks. I'll post a link later, when it is available.

Ali Enayat and I are currently preparing a paper whose main theorem is that with respect to definable classes, Ord is NOT weakly compact. In particular, we show, in every model of ZFC,

• there is a definable Ord-tree with no definable cofinal branch.
• there is a definable 2-coloring of $V$, with no definable homogeneous proper class.
• there is a definable set-satisfiable $L_{\text{Ord},\omega}$-theory, which has no definable class model.

This result surprised me very much, since it shows that with respect to definable classes, we can prove that Ord fails to have a large cardinal property that reasonable people might have expected to hold true.

The preprint is not ready yet, but I expect that it will be ready in a few weeks. I'll post a link later, when it is available.

Ali Enayat and I are currently preparing a paper whose main theorem is that with respect to definable classes, Ord is NOT weakly compact. In particular, we show, in every model of ZFC,

• there is a definable Ord-tree with no definable cofinal branch.
• there is a definable 2-coloring of $V$, with no definable homogeneous set.
• there is a definable set-satisfiable $L_{\text{Ord},\omega}$-theory, which has no definable class model.

This result surprised me very much, since it shows that with respect to definable classes, we can prove that Ord fails to have a large cardinal that reasonable people might have expected to hold true.

The preprint is not ready yet, but I expect that it will be ready in a few weeks. I'll post a link later, when it is available.

Ali Enayat and I are currently preparing a paper whose main theorem is that with respect to definable classes, Ord is NOT weakly compact. In particular, we show, in every model of ZFC,

• there is a definable Ord-tree with no definable cofinal branch.
• there is a definable 2-coloring of $V$, with no definable homogeneous proper class.
• there is a definable set-satisfiable $L_{\text{Ord},\omega}$-theory, which has no definable class model.

This result surprised me very much, since it shows that with respect to definable classes, we can prove that Ord fails to have a large cardinal that reasonable people might have expected to hold true.

The preprint is not ready yet, but I expect that it will be ready in a few weeks. I'll post a link later, when it is available.