Julian Fernandez
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Registered User
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I'm a physicist
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Mar 30 |
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Cardinality of classes Do this "largest" cardinal have a name? |
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Mar 30 |
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Cardinality of classes But, if we extend the definition of cardinal to bijections among elements of two entities (not necessarily sets, but also proper classes), why can't we define the cardinality of the proper class of the set of all sets? It would be useful in the sense that now you do have the largest possible cardinal number (it would not be inconsistent with cantor's theorem because the proper class in not a set so there is not such thing a the power set of a class). Why is this not useful? |
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Mar 27 |
awarded | ● Commentator |
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Mar 27 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? $\omega_1$ many thanks!!! |
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Mar 27 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? Please let me know if you don't want me to keep posting questions to you here. My understanding was that the machine reaches stage $\omega_1$ in finite time, but according to an answer to a question that I posted at math.se ( math.stackexchange.com/questions/341895/… peo) it should take infinite time to reach stage $\omega_1$. Did I misunderstood you article? |
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Mar 26 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? Thanks again! ... |
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Mar 26 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? Thanks, I'll read about their work. But I was just curious about your personal opinion, which I highly valuate, of why you restricted yourself in the initial proposal to the time generalization but to the space one. Did you considered it but didn't like it, or you just didn't considered it at all? |
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Mar 26 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? Thanks! just one more question (or should I post it a a separate question?) Is there any reason you extended the classical concept into transfinite ordinal time but not into transfinite ordinal space? |
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Mar 26 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? @Joel Is the fact that $\Delta_2^1$ is closed under the boldface jump $A^\blacktriangledown$ due to the fact that you limited the oracles to questions about writable (or constructible) reals (which is limited by the complexity of the limit stage rules), instead of allowing an extra tape with uncountable number of cells (which as you mention, clearly we cannot expect always to be able to write such an object out on a tape, but suppose you could). So your oracles are not as strong (arbitrary sets of reals) as they could potentially be? |
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Feb 8 |
awarded | ● Scholar |
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Feb 8 |
awarded | ● Supporter |
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Feb 8 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? Joel, super clear answer, case closed. Thanks!!! |
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Feb 8 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? Joel, yes, actually, the reason I chose a $\Sigma_1^2$ is that I was interested to know if it could decide the CH (which I understand you do not think has a specific answer, based on your multiverse theory) |
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Feb 8 |
awarded | ● Editor |
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Feb 8 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? actually I used "natural" instead of "real", I already corrected it. Thanks! |
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Feb 8 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? changed natural for real, which was wrong |
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Feb 8 |
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Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? no, I mean of real numbers, because the input of his machines is a real number, that is, an infinite sequence of 1's and 0's (the tape has infinite length). this is an extract from his article: Thus,we want somehow to allow a set of real numbers,such as the halting problem H,to become an oracle for the machines.And since such a set could be uncountable,and these in particular is definitely uncountable,we can’t expect to be able to write out the entire contents of the oracle on an extra tape,as in the classical theory." |
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Feb 8 |
awarded | ● Student |
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Feb 8 |
asked | Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? |
