bio  website  jdh.hamkins.org 

location  New York City  
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I am a professor at the City University of New York, at the College of Staten Island and the CUNY Graduate Center, living in midtown Manhattan. My main research interest lies in mathematical logic, particularly set theory, focusing on the mathematics and philosophy of the infinite. A principal concern has been the interaction of forcing and large cardinals, two central concepts in set theory. I have worked in group theory and its interaction with set theory in the automorphism tower problem, and in computability theory, particularly the infinitary theory of infinite time Turing machines. Recently, I am preoccupied with the settheoretic multiverse, engaging with the emerging field known as the philosophy of set theory.
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What is the maximal number of distinct values of the product of n permuted ordinals
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answered  What is the maximal number of distinct values of the product of n permuted ordinals 
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What does a Turing machine compute?
See related question: mathoverflow.net/q/141104/1946 
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Rationale behind an requirement on Turing machines
Of course one can still expect to compute some things, as you've indicated. But your machines are not suitable for a fullfledged theory of computability, since they are not Turing complete. We could also handicap other kinds of Turing machines in various ways, to prevent them from being Turing complete, such as by insisting they run in polynomial time. Such a polytime handicap offers a computational model of an extremely robust and important class of decidability problems. Your model similarly handles some class of decidability problems, but I'm not sure how robust it is. 
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Rationale behind an requirement on Turing machines
I understood the situation as: you write the input on the tape (in binary) and then fill up the rest of the tape with blanks, that is, with $0$s. 
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Rationale behind an requirement on Turing machines
It is fine for MO, in my opinion. 
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Rationale behind an requirement on Turing machines
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Rationale behind an requirement on Turing machines
Yes. For example, one could use unary input, which amounts in this case to imposing the Hopcroft/Ulman requirement. 
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answered  Rationale behind an requirement on Turing machines 
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Exponentiation and Dedekindfinite cardinals
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answered  Exponentiation and Dedekindfinite cardinals 
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dual (p,q)property
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dual (p,q)property
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answered  dual (p,q)property 
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dual (p,q)property
By a "setsystem", you meant just that $S$ is a collection of subsets of $X$? 
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What are the most attractive Turing undecidable problems in mathematics?
@ChristophSimonSenjak Yes, there are such intermediate problems, strictly between the decidable sets and the halting problem. There is a c.e. problem that is not decidable, but which is not Turingequivalent to the halting problem. This is the solution of Post's problem, solved by FriedburgMuchnik. See en.wikipedia.org/wiki/…. 
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Can there ever be symbolic formalism without intuitive heuristics?
My point was that perhaps you haven't succeeded in asking the question that you meant to ask, since the actual question you asked seems to have this easy affirmative answer, namely, Yes, we can create a symbolic formalism without starting from an intuitive seed. 
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Can there ever be symbolic formalism without intuitive heuristics?
Isn't it easy to create a symbolic formalism without an intuitive seed? One can just make up some syntactic rules, even randomly. For example, off the top of my head: form strings of symbols by starting with the empty string, and then appending either $000101$ to any string, or if it ends in $01$, then replacing that with $11101$; and if $1111$ occurs as a substring, replace with $1$. This seems to be a formal system without any intuitive basis. 