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Results tagged with lo.logic
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user 13530
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
4
votes
Set theory without the empty set
I’ve explored a “set theory whose axioms do not prove the existence of an empty set,” the Incomprehensive Set Theory in my “Naive View of the Russell Paradox” (https://arxiv.org/abs/2103.00090), but f …
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2
votes
Can we have a theory that define small (ZFC set sizes) collections of big sets?
Church’s Set Theory with a Universal Set, and my variant of it, though not Mitchell’s† can do some of what you ask, e.g. schema of Replacement from well founded sets, but not, I think, the set of all …
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4
votes
Accepted
Can we add set complements on top of ZF?
Church’s “Set Theory with a Universal Set” and its variants have complements, with Replacement restricted to sets equinumerous to a well-founded set:
Alonzo Church 1974a. “Set Theory with a Universa …
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4
votes
Accepted
What does the axiom of replacement mean and why should I believe it?
For an argument that the iterative conception implies something weaker than unrestricted Separation (implied by unrestricted Replacement), i.e. $\Sigma_2$ Replacement, see Randall Holmes 2001 http://m …
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1
vote
Why should we believe in the axiom of regularity?
You’re not obliged to believe the axiom; see, for instance, https://en.wikipedia.org/wiki/Universal_set.
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5
votes
Where is the end of universe?
Have a look at Church’s (first) Set Theory with a Universal Set, which is equiconsistent with ZFGC, though Church didn’t publish a full proof, and I think he abandoned the proof in his archives at Pri …
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5
votes
What would be some major consequences of the inconsistency of ZFC?
(This began as a comment to Henry Cohn’s answer, which which I mostly agree, but I’m over the character limit.) I think the specifics of the example in case 4 actually overlap with 1; an easily-remedi …
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5
votes
Can we have A={A} ?
(This is intended as a comment on Thomas Forster’s point above, “i think there is a literature about them that goes back earlier than Quine,” but I’m over the character limit.)
Much earlier than …
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1
vote
mutually incompatible abstraction terms?
For a philosophical perspective on this, see Alan Weir’s “Neo-Fregeanism: An Embarrassment of Riches”: “The embarrassment of riches objection is that there is a plurality of consistent but pairwise in …
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3
votes
Physics and Church–Turing Thesis
To follow up on Michael Beeson’s answer (I’m not allowed to post comments yet), Robin Gandy’s paper was “Church's Thesis and Principles for Mechanisms” in The Kleene Symposium (http://dx.doi.org/10.10 …
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5
votes
How much of ZFC does Quine's New Foundations prove?
While I second (with highly biased motivation) the recommendation of Forster’s book, for questions like this an easier starting point than Forster’s book or Holmes’ articles might be Holmes’ book Elem …
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