************************************************* The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture). There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see this blog comment. However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture. ************************************************ (DeepL翻訳) (IUTT-IV)の最後の部分では、ZFC集合論の異なるモデルを用いて、より完全に普遍的なテイヒミュラー理論を発展させることを検討しています(この部分はabc推論への応用には必要ありません)。 この論文の第3節43ページのGrothendieck宇宙公理の追加によるZFCの拡張の保守性に関する発言には不正確な点があるようですが、このブログのコメントを参照してください。 しかし、この発言は純粋に動機付けのためのものであり、abcの予想の証明に影響を与えるものではありません。 0534132人目の素数さん2020/05/09(土) 13:52:51.91ID:bijIFi3Y>>533 何が決着したのかはよく分からないが、
1.下記の「断り書き」を追加 ←あまり重要ではない。 ・【最終版頁67行2】" albeit from an extremely naive/non-expert point of view, relative to the theory of foundations! " ・【最終版頁67行6】"[i.e., mathematicians who are not equipped with a detailed knowledge of the theory of foundations!]" ・【最終版頁68行7】”in the context of the present series of papers" 0580132人目の素数さん2020/05/09(土) 16:57:50.01ID:/BYRDNlz 2.おそらくタオ指摘で削除した部分 【最終版頁68行10】 ← 重要 ・Although we shall not discuss in detail here the quite difficult issue of whether or not there actually exist ZFCG-models, we remark in passing that one may justify the stance of ignoring such issues — at least from the point of view of establishing the validity of various“final results”that may be formulated in ZFC-models -の後、
・投稿版:by invoking a result of Feferman [cf. [Ffmn],§2.3] concerning the “conservative extensionality” of ZFCG relative to ZFC, i.e., roughly speaking, that“any proposition that may be formulated in a ZFC-model and, moreover, holds in a ZFCG-model infact holds in the original ZFC-model”. ・最終版:by invoking the work of Feferman [cf. [Ffmn]]. Precise statements concerning such issues, however, lie beyond the scope of the present paper [as well as of the level of expertise of the author!].
3.「言いたかったこと」を言い直した追記 【最終版頁68行15】 , with the following proviso: In the following discussion, it should be understood that every set-theoretic formula that appears is “absolute” in the sense that its validity for a collection of sets contained in some universe V relative to the model of set theory determined by V isequivalent, for any universe W such that V ∈ W , to its validity for the same collection of sets relative to the model of set theory determined by W [cf., e.g., [Drk], Chapter 3, Definition 4.
こんな言葉で非ユークリッド幾何や相対性理論が否定できるなら苦労しない 0582132人目の素数さん2020/05/09(土) 16:59:21.31ID:/gPwFwIn>>579 >albeit from an extremely naive/non-expert point of view, relative to the theory of foundations!
IUT-4で以下を修正しており、指摘で修正が必要であったのは、この箇所だよ。 concerning the “conservative extensionality” of ZFCG relative to ZFC, i.e., roughly speaking, that“any proposition that may be formulated in a ZFC-model and, moreover, holds in a ZFCG-model infact holds in the original ZFC-model”
>ZFCの9個の公理 のIUT-4の記載部分は、投稿版も最終版も、以下記載だったが、修正されてないので此処は問題でないよ。 ”[i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].”