[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む79 (1002ﾚｽ)
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932(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2020/01/03(金) 19:59:33.09 ID:ivt0JCXh(26/37) AAS
>>931
つづき
４）
ZFCG-modelを考えたけど、ギブアップしたのかな（＾＾；
P67
Section 3: Inter-universal Formalism: the Language of Species
(†G) Given any set x, there exists a universe V such that x ∈ V .
We shall refer to a ZFC-model that also satisfies this additional axiom of the
Grothendieck school as a ZFCG-model.
P68
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 it may be
possible to justify the stance of ignoring such issues in the context of the present
series of papers ? at least from the point of view of establishing the validity of
various “final results” that may be formulated in ZFC-models ? 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!].
In the following discussion, we use the phrase “set-theoretic formula” as it is
conventionally used in discussions of axiomatic set theory [cf., e.g., [Drk], Chapter 1,
§2], 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 is equivalent, 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 4.2].
Remark 3.1.2.
(ii) One interesting point of view that arose in discussions between the author
and F. Kato is the following. The relationship between the classical approach to
discussing mathematics relative to a fixed model of set theory ? an approach in
つづく
933(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2020/01/03(金) 20:01:30.16 ID:ivt0JCXh(27/37) AAS
>>932
つづき
５）
a [small] category は、集合論ZFC内の speciesでやれる？
P71
Remark 3.1.4. Note that because the data involved in a species is given by
abstract set-theoretic formulas, the mathematical notion constituted by the species
is immune to, i.e., unaffected by, extensions of the universe ? i.e., such as
the ascending chain V0 ∈ V1 ∈ V2 ∈ V3 ∈ ... ∈ Vn ∈ ... ∈ V that appears in
the discussion preceding Definition 3.1 ? in which one works. This is the sense
in which we apply the term “inter-universal”. That is to say, “inter-universal
geometry” allows one to relate the “geometries” that occur in distinct universes.
P72
Example 3.2. Categories. The notions of a [small] category and an isomorphism class of [covariant] functors between two given [small] categories yield an
example of a species. That is to say, at a set-theoretic level, one may think of a
[small] category as, for instance, a set of arrows, together with a set of composition
relations, that satisfies certain properties; one may think of a [covariant] functor
between [small] categories as the set given by the graph of the map on arrows determined by the functor [which satisfies certain properties]; one may think of an
isomorphism class of functors as a collection of such graphs, i.e., the graphs determined by the functors in the isomorphism class, which satisfies certain properties.
Then one has “dictionaries”
0-species ←→ the notion of a category
1-species ←→ the notion of an isomorphism class of functors
at the level of notions and
a 0-specimen ←→ a particular [small] category
a 1-specimen ←→ a particular isomorphism class of functors
at the level of specific mathematical objects in a specific ZFC-model. Moreover, one
verifies easily that species-isomorphisms between 0-species correspond to isomorphism classes of equivalences of categories in the usual sense.
つづく
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