Contribution to the theory of semisets VI: (Non-existence of the class of all absolute natural numbers)
✍ Scribed by Antonin Sochor
- Publisher
- John Wiley and Sons
- Year
- 1975
- Tongue
- English
- Weight
- 299 KB
- Volume
- 21
- Category
- Article
- ISSN
- 0044-3050
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✦ Synopsis
Z e a c h r . i. d. L & k und Qrundlagen d . A ¶&. Hd. P I , S. 439-412 (1975) CONTRIBUTION TO THE THEORY OF SEMISETS VI (Non-existence of the class of all absolute natural numbers) by ANTONIT SOCHOR in Prague (C.S.S.R.)
.4n important concept-the class of all absolute natural numbers has been introduced in the theory of semisets (TSS). Recent development of this theory has shown that this class may play an important role, e.g. if we investigate ROBINSON'S nonstandard analysis in TSS. The class of all absolute natural numbers (denoted by An) is tlw class the members of which are exactly natural numbers containing no proper sciiiisc~t (i.e. a class which is not a set and which is a part of a set). Formally,
[3]). We can also say (cf. C 615)l) t h a t An is thc minimal complete proper subsemiset of wo. A natural question arises whether thc chxistcncc of AII is provable in TSS. This paper shows that the answer to the questiori is negative, since wc shall construct n syntactic model of TSS in which An does not c b s i s t .
To prove this result we shall construct, some kind of ultraproduct relation R and w .slid1 intcrprete classes as certain subclasses of W(R) and E as usual (cf. C 607, C 612; sets being interpreted as all classes of thc form R"(z)). If we denote all concepts involved in our interpretation by * then the following will be provable:
(1) All GODEL'S axioms except C 4 (and moreover (A 2)-(A 7), (C 2) from [l]) in
(2) Kon-standardness (i.e. (3X) (X g (u,& i M ( X ) ) holds in the model).
(3) An does not exist in * sense (i.c. among *-classes which art. not *-sets there is A t fiist let us formulate the idea of the proof: We shall construct relations R, for ever). 11 ECO, such that R = U It,, and such that R,+l is isomorphic to ultrapower of It,, on i'(oo) (see
and moreover also to every (saturated) part X of C(R,) there is a corresponding part S(X, n , 71 + 1) of C(R,l+l) and tl corresponding part S ( X , n) of C(R) (see C 607). This corrwpondence is an elementary embedding (see C 611). Now, *-classes will be classes of the form S ( X , n) for some (saturated) part X of C(R,) (see C 612). Evidently, these *-classes form a model of TSS (see C 612). We can show that no *-class is the class of all absolute natural numbers in the model. Suppose on the contrary that for a part X of C(R,,) H'C have S ( X , n) = An*. It is easy to prove that the (n + 1)-th diagonal is an elemtmt of S ( X , n) and moreover this diagonal contains S( Y , n + 1) where Y means An in ultrapower It,,,, (i.c. the class determined by all constants of natural numbers) (see C614). * -scnsc. i i o minimal completc part of w,*>. newO I ) Kumbera consisting of four digits refer to items in [l]; analogically a reference beginning with Cj' refers to the j-th contribution to the theory of eemiaets.