This offering is quite old, and I have since attempted what may be a better line of attack. Might be on the Footnotes to Plato blog.
The following proof of the Schroeder-Bernstein theorem is, as far as I know, new as of April 2006. Notation: "cardA" denotes the cardinal number of a set A. "<=" means "less than or equal to." "~" means "equinumerous to."
Schroeder-Bernstein theorem: If, and only if, A and B are sets such that card A <= cardB and card B <= card A, then cardA = cardB.Proof:
Remark: We grant that the transitivity of subsets is established from axioms. That is, A Í B Í C implies A Í C. Further, for nonempty sets, we accept that if A Ì B, then B Ì A is false. This follows from the definition of subset: for all x, x element of X implies x element of Y. But, if A is a proper subset of A, then, for some b, b element of B implies b is not an element of A. Hence B does not fit the definition of subset of A.
We rewrite cardA <= cardB and card B <= card A, thus:
A ~ C Í B and B ~ D Í A
If
Suppose A ~ C = B. Then we are done. Likewise for B ~ D = A. So we test the situation for nonempty sets with: A ~ C Ì B and B ~ D Ì A. (We note that if C ~ D, then A ~ B, spoiling our interim hypothesis. So we assume C is not equinumerous to D.)
We now form the sets A U C and B U D, permitting us to write (A U C) Ì (A U B) and (B U D) Ì (A U B).
We now have two options:
i) (A U C) Ì (A U B) Ì (B U D) Ì (A U B)
This is a contradiction since A U B cannot be a proper subset of itself.
ii) (A U C) Ì (A U B) É (B U D) Ì (A U B)
Also a contradiction, since B U D cannot be a proper subset of A U B and also properly contain A U B.
Only if
We let B = C and D = A.
This proof, like the standard modern proof, does not rely on the Axiom of Choice.
Alternate proof
of the Schroeder-Bernstein theorem
Draft 2The following proof of the Schroeder-Bernstein theorem is, as far as I know, new as of April 2006. Notation: "cardA" denotes the cardinal number of a set A. "<=" means "less than or equal to." "~" means "equinumerous to."
Schroeder-Bernstein theorem: If, and only if, A and B are sets such that card A <= cardB and card B <= card A, then cardA = cardB.Proof:
Remark: We grant that the transitivity of subsets is established from axioms. That is, A Í B Í C implies A Í C. Further, for nonempty sets, we accept that if A Ì B, then B Ì A is false. This follows from the definition of subset: for all x, x element of X implies x element of Y. But, if A is a proper subset of A, then, for some b, b element of B implies b is not an element of A. Hence B does not fit the definition of subset of A.
We rewrite cardA <= cardB and card B <= card A, thus:
A ~ C Í B and B ~ D Í A
If
Suppose A ~ C = B. Then we are done. Likewise for B ~ D = A. So we test the situation for nonempty sets with: A ~ C Ì B and B ~ D Ì A. (We note that if C ~ D, then A ~ B, spoiling our interim hypothesis. So we assume C is not equinumerous to D.)
We now form the sets A U C and B U D, permitting us to write (A U C) Ì (A U B) and (B U D) Ì (A U B).
We now have two options:
i) (A U C) Ì (A U B) Ì (B U D) Ì (A U B)
This is a contradiction since A U B cannot be a proper subset of itself.
ii) (A U C) Ì (A U B) É (B U D) Ì (A U B)
Also a contradiction, since B U D cannot be a proper subset of A U B and also properly contain A U B.
Only if
We let B = C and D = A.
This proof, like the standard modern proof, does not rely on the Axiom of Choice.
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