Inform speaking ontological optimisms means that everything that possibly exists in the abstract reality actually exists. From this principle we (again informally) get the Axiom of infinity, the Power set axiom. the Axiom of choice and the negation of the continuum hypothesis. How can we formalize ontological optimism? I think about something like a basic set theory $B$ and an axiom like: If $B\wedge \exists M\in Set:F(M)$ and $B\wedge\neg\exists M\in Set:F(M)$ are equiconsistent than $\exists M\in Set:F(M)$. But I do not see in which logical framework I may formulate such an axiom.
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asked Aug 20, 2019 at 18:15
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9$\begingroup$ What about the two sentences, "There is a bijection between $\mathbb R$ and $\omega_1$," and "There is a bijection between $\mathbb R$ and $\omega_2$"? Which one of these possible things is optimistic? $\endgroup$– Monroe EskewCommented Aug 20, 2019 at 21:08
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5$\begingroup$ Why do we get the axiom of choice? What about "There is a non-well-orderable set?" $\endgroup$– Noah SchweberCommented Aug 20, 2019 at 22:32
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5$\begingroup$ That said, you might be interested in the inner model hypothesis, which in my opinion is much underrated as a foundational perspective. $\endgroup$– Noah SchweberCommented Aug 20, 2019 at 23:56
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2$\begingroup$ Looking at the informal statement "everything that possibly exists in the abstract reality actually exists": if we interpret "possibly exists" to mean "exists in a generic extension" (which I understand is not quite what you mean) then a way to avoid the problem pointed out by Monroe is to strengthen the hypothesis of "possibly exists" to "possibly necessarily exists", meaning that once forced to be true, it cannot subsequently be forced to be false in a further generic extension. The resulting statement is known as the Maximality Principle. $\endgroup$– Trevor WilsonCommented Aug 24, 2019 at 14:35
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3$\begingroup$ @TrevorWilson The intuitive idea in the question seems to point toward the maximality principle with arbitrary parameters, which (as Hamkins points out in the paper you linked to) contradicts ZF (and even Z) by implying that all sets are countable. But that might be where ontological optimism naturally leads. $\endgroup$– Andreas BlassCommented Aug 28, 2019 at 15:20
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As remarked above, claims from "optimism" are not very clear-cut: e.g. do we think of choice as asserting the existence of well-orderings of arbitrary sets, or the non-existence of cool stuff like amorphous sets? In particular, insofar as AC is optimistic it seems to me that CH should be optimistic for the same reason (each asserts that every set, or every set of the appropriate type, admits a certain kind of structure - a well-ordering or a bijection with $\mathbb{R}$). I personally view optimism of this kind as fundamentally un-formalizable.
However, there is a set-theoretic principle which in my opinion takes a very interesting stab in this direction: the inner model hypothesis (and its variants). Roughly speaking, IMH says that a lot of stuff we could add to the set-theoretic universe is already there. In particular, it implies the negation of AC. That said, it also has some consequences which are arguably pessimistic, namely it implies the nonexistence of inaccessible cardinals.
answered Oct 2, 2019 at 15:38