ガロア第一論文と乗数イデアル他関連資料スレ13

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950: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/02/15(土) 09:56:20.09 ID:XknlDm4+

>>945 補足
>A proof that Zorn's lemma implies the axiom of choice illustrates a typical application of Zorn's lemma.[17]

えーと、最後の [17]を見ると下記だ
Notes
17  Halmos 1960, § 16. Exercise.
References
Halmos, Paul (1960). Naive Set Theory. Princeton, New Jersey: D. Van Nostrand Company.
https://en.wikipedia.org/wiki/Naive_Set_Theory_(book)
Naive Set Theory (book)

うーんと、海賊版を探すと
Naive set theory.
Halmos, Paul R. (Paul Richard), 1916-2006.
Princeton, N.J., Van Nostrand, [1960]

があった (下記 文字化けと乱丁ご容赦)
Sec. 16  ZORN'S LEMMA p65
Exercise.
Zorn's lemma is equivalent to the axiom of choice.
[Hint
for the proof: given a set X, consider functions /such that dom/C
(P(X), ran/dX, and f(A)eA for all A in dom/; order these functions
by extension, use Zorn's lemma to find a maximal one among them, and
prove that if/ismaximal, then dom/= <P(X)
—
{0}.] Consider each
of the following statements and prove that they too are equivalent to
the axiom of choice.
(i)
Every partially ordered set has a maximal
chain (i.e., a chain that
is
not
a
proper subset of any other chain).
(ii)
Every chain in
a
partially ordered set
is
included in some maximal chain.
(iii) Every partially ordered set in which each chain has
a
least upper
bound has a maximal element.
(引用終り)

か
解答はないかな？・・・ ないね・・  ；ｐ）

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