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Jan
28
comment Taller models of ZFC
@Asaf Karagila: I think you are thinking about the consistency of the statement (you are saying it is consistent relative to the existence of one inaccessible), but I was thinking about the provability of the statement. Is it consistent with ZFC that the collection of wordly cardinals is nonempty and bounded? In my earlier comment, I was trying to suggest that the axiom would follow from very mild large cardinal assumptions.
Jan
28
comment Taller models of ZFC
In (2), are you referring to transitive set models (so that the question is interpreted within a single class model of ZFC, so to speak) or are you asking about class models (so that the question is about the relationship between different class models of ZFC)? In the former case, doesn't (2) follow from the axiom that every set is contained in a transitive set model of ZFC, which in turn follows just from the existence of arbitrarily large inaccessible cardinals?
Jan
26
comment Is platonism regarding arithmetic consistent with the multiverse view in set theory?
I noticed this because I had originally hoped that the multiverse perspective would support position $P$, but I just don't see it after looking at the paper.
Jan
26
comment Is platonism regarding arithmetic consistent with the multiverse view in set theory?
@Nik Weaver: yes, I don't think the multiverse view is especially helpful for the position (which is very sympathetic) that $\omega$ has some real meaning but $2^\omega$ does not. Let's call that position $P$. The paper by Gitman and Hamkins has a list of multiverse axioms, including that the $\omega$ of any model is ill-founded in some other model. I don't see an obvious justification why one would accept all the other multiverse axioms of that paper, but not accept that particular multiverse axiom. It seems that position $P$ in the multiverse view would require such a justification.
Jan
22
comment Is platonism regarding arithmetic consistent with the multiverse view in set theory?
It seems to me that one straightforward answer is that, according to a theorem of Gitman and Hamkins, one of the axioms consistent with the "multiverse view" is that every model of set theory is ill-founded with respect to some other model. If this axiom holds, then the concept of a unique "standard" model of arithmetic becomes very doubtful. See "A natural model of the multiverse axioms", projecteuclid.org/euclid.ndjfl/1285765800
Jan
19
comment Constructive compactness for countable models?
@katz: I would find that quite surprising. One common aspect of constructive mathematics is an avoidance of formalization, and of formal systems in general. Sometimes logicians like myself will impose a particular axiomatic framework on BISH, but strictly speaking this is a departure from the intentions of the authors. Even in constructive reverse mathematics the base system is often unspecified.
Jan
18
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Jan
18
comment Constructive compactness for countable models?
@Noah Schweber: in a professional context of logic, I would read "constructive" to mean something like "in intuitionistic logic", rather than the weaker informal meaning that some mathematicians use for it (e.g. "provable in ZF").
Jan
18
comment Constructive compactness for countable models?
The compactness theorem is a classical theorem about classical models. It would be very unusual for someone in constructive mathematics to worry about such a result. On one hand, constructivists such as Bishop avoid formalization entirely, and thus also avoid model theory. Bishop's interest, essentially, is core math only. On the other hand, constructive metamathematics is done with alternative kinds of models that are relevant to constructive logic. The concept of a model (in the sense of classical model theory) has classical logic through and through (e.g. in the T-schema).
Jan
16
comment Compactness for countable models?
That makes sense. @Noah
Jan
16
revised Compactness for countable models?
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Jan
16
revised Compactness for countable models?
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Jan
16
revised Compactness for countable models?
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Jan
16
answered Compactness for countable models?
Jan
11
comment Papers that debunk common myths in the history of mathematics
Cf. this answer on Math.StackExchange.com: math.stackexchange.com/a/845583
Jan
10
comment What is the modern consensus on the difficulty of infinitesimals?
@katz: Please see my previous comment. Regarding this question, the usual way we would look at nonstandard analysis in Reverse Mathematics is to look at conservation theorems which say "if a theorem of such and such syntactic form is provable in this particular system of axioms for nonstandard analysis then the theorem is also provable in that particular standard system". Many results have been obtained. We don't normally look at theorems which say "a model of such and such system of nonstandard analysis exists", for various reasons.
Jan
10
comment What is the modern consensus on the difficulty of infinitesimals?
So much commentary might have been avoided by simply sending me an email! I left a follow-up comment on Jan. 7 at math.stackexchange.com/questions/1602977/… which explains what I mean. // I think the phrase "infinitesimal number" has some ambiguity. It seems possible to give a (more or less) concrete example of an ordered field containing infinitesimals and "infinite" elements. However, it isn't clear that such objects are infinite numbers - is the term "number" often used for an element of an arbitrary field?
Dec
29
revised What are some important but still unsolved problems in mathematical logic?
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Dec
29
answered What are some important but still unsolved problems in mathematical logic?
Dec
23
comment Do mathematical objects disappear?
The notion of "infinitesimally small numbers" itself disappeared, eventually. It is now studied in nonstandard analysis, which as the name suggests is different from the "standard" approach used in calculus and real analysis.