Frege observes that this is the case if and only if (henceforth: iff) the concepts are

equinumerous in the sense that the objects falling under one concept can be one-one correlated to the objects falling under the other.

For a variety of sequences s, natural statistics on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] have the same distribution as natural statistics on other

equinumerous combinatorial families, a phenomenon that was used to settle an open question about Coxeter groups in [21].

Notice that this last proof shows that the three sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (321, 231), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](321, 312) and[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](321,231,312) are identical, not just

equinumerous. To my knowledge, there is no analogous result in the enumeration of general (as opposed to irreducible) permutations.

The training dataset was randomly divided into ten distinct

equinumerous subsets, for the purpose of the k-fold method.

Compare this: when it seems to us that a set could not possibly be

equinumerous with a proper subset, this seeming has no apparent correlated experience distinct from it.

To see that Cantor did not intend any restriction to definable sets, one only has to consider his remarks on the "internal determinateness" of the question whether two sets are

equinumerous. (70) From the discussion it is clear that the "law" by which elements of the two sets are put in correspondence need not be definable.

(16.) This leads to the familiar proofs that the set of even numbers is

equinumerous with the set of odd numbers, and that the natural numbers are

equinumerous with the rationals.

Part (2) of the BusyCode-Motion Lemma shows that [f.sub.CM,p]is onto; by the computational optimality of BCM (OCM Theorem 3.13), the domain and range of [f.sub.CM,p] are

equinumerous, so it must be one-to-one as well.

Yet again, two stroke string inscriptions are inscriptions of the same stroke string type if and only if the strokes which make up the inscriptions are

equinumerous. But if the imagined inscriptions consist of no definite number of strokes such that we could tell what that number is, it is uncertain what an

equinumerous relation here could come to.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is shown to be

equinumerous to such paths with [??] = 2 (Theorem 2.4), and more generally [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is

equinumerous to such paths with [??] = k (Theorem 2.3).

Clearly the members of either set not containing an occurrence of 113 or 133, respectively, are

equinumerous by the preceding.

He can, however, be credited with the following: he derived (or showed how to derive) the whole of arithmetic in second-order logic from "Hume's principle" ("the numbers belonging to F and G are equal if and only if F is

equinumerous with G"); he derived Peano's second postulate ("every natural number has a successor") from the same principle; he succeeded, where Dedekind failed, in demonstrating the existence of an infinite system (p.