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The partial order of statements $\leq$ the axiom of choice is simply the lower-cone below AC in the Lindenbaum algebra of ZF statements. Thus, in particular, it is a Boolean algebra, where in this case, the equivalence class of AC is $1$. With this perspective, one quickly realizes that every nontrivial statement $\varphi$ strictly below AC forms a maximal antichain with its relative complement $\text{AC}\wedge\neg\varphi$. So of course, every nontrivial statement is part of a finite maximal antichain. (Asaf had noted a few instances of this in his answer.)

Meanwhile, on the other side of the question, let me mention that since it is known, assuming the consistency of ZF, that there are infinitely many inequivalent statements weaker than AC (such as choice for families of various-sized finite sets, or DC for increasingly long transfinite sequences), it follows that the Boolean algebra is infinite. From this, it follows that there must be infinite antichains, simply because every infinite Boolean algebra has an infinite antichain.

The partial order of statements weaker than the axiom of choice is simply the upper-cone above AC in the Lindenbaum algebra of ZF statements. That is, it is a quotient of the Lindenbaum algebra by the principal ideal generated by the statement AC. In particular, it is a Boolean algebra, where in this case, the equivalence class of AC is $0$. With this perspective, one quickly realizes that every nontrivial statement $\varphi$ strictly weaker than AC, and therefore strictly above AC in the Lindenbaum algebra, forms a maximal antichain with its relative complement $\text{AC}\vee\neg\varphi$. So of course, every nontrivial statement is part of a finite maximal antichain. (Asaf had noted a few instances of this in his answer.)

Meanwhile, on the other side of the question, let me mention that since it is known, assuming the consistency of ZF, that there are infinitely many inequivalent statements weaker than AC (such as choice for families of various-sized finite sets, or DC for increasingly long transfinite sequences), it follows that the Boolean algebra is infinite. From this, it follows that there must be infinite antichains, simply because every infinite Boolean algebra has an infinite antichain.

The partial order of statements $\leq$ the axiom of choice is simply the lower-cone below AC in the Lindenbaum algebra of ZF statements. Thus, in particular, it is a Boolean algebra, where in this case, the equivalence class of AC is $1$. It follows that every nontrivial statement $\varphi$ that is strictly below AC forms a maximal antichain with its relative complement $\text{AC}\wedge\neg\varphi$. So of course, every nontrivial statement is part of a finite maximal antichain. (Asaf had noted a few instances of this in his answer.)

Meanwhile, since it is known, assuming the consistency of ZF, that there are infinitely many inequivalent statements weaker than AC (such as choice for families of various-sized finite sets, or DC for increasingly long transfinite sequences), it follows that the Boolean algebra is infinite. From this, it follows that there must be infinite antichains, simply because every infinite Boolean algebra has an infinite antichain.

The partial order of statements $\leq$ the axiom of choice is simply the lower-cone below AC in the Lindenbaum algebra of ZF statements. Thus, in particular, it is a Boolean algebra, where in this case, the equivalence class of AC is $1$. With this perspective, one quickly realizes that every nontrivial statement $\varphi$ strictly below AC forms a maximal antichain with its relative complement $\text{AC}\wedge\neg\varphi$. So of course, every nontrivial statement is part of a finite maximal antichain. (Asaf had noted a few instances of this in his answer.)

Meanwhile, on the other side of the question, let me mention that since it is known, assuming the consistency of ZF, that there are infinitely many inequivalent statements weaker than AC (such as choice for families of various-sized finite sets, or DC for increasingly long transfinite sequences), it follows that the Boolean algebra is infinite. From this, it follows that there must be infinite antichains, simply because every infinite Boolean algebra has an infinite antichain.

The partial order of statements $\leq$ the axiom of choice is simply the lower-cone below AC in the Lindenbaum algebra of ZF statements. Thus, in particular, it is a Boolean algebra, where in this case, the equivalence class of AC is $1$. It follows that every nontrivial statement $\varphi$ that is strictly below AC forms a maximal antichain with its relative complement $\text{AC}\wedge\neg\varphi$. So of course, every statement is part of a finite maximal antichain. (Asaf had noted a few instances of this in his answer.)

Meanwhile, since it is known, assuming the consistency of ZF, that there are infinitely many inequivalent statements weaker than AC (such as choice for families of various-sized finite sets, or DC for increasingly long transfinite sequences), it follows that there must be infinite antichains, because every infinite Boolean algebra has an infinite antichain.

The partial order of statements $\leq$ the axiom of choice is simply the lower-cone below AC in the Lindenbaum algebra of ZF statements. Thus, in particular, it is a Boolean algebra, where in this case, the equivalence class of AC is $1$. It follows that every nontrivial statement $\varphi$ that is strictly below AC forms a maximal antichain with its relative complement $\text{AC}\wedge\neg\varphi$. So of course, every nontrivial statement is part of a finite maximal antichain. (Asaf had noted a few instances of this in his answer.)

Meanwhile, since it is known, assuming the consistency of ZF, that there are infinitely many inequivalent statements weaker than AC (such as choice for families of various-sized finite sets, or DC for increasingly long transfinite sequences), it follows that the Boolean algebra is infinite. From this, it follows that there must be infinite antichains, simply because every infinite Boolean algebra has an infinite antichain.