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May 18th, 2015, 10:17 AM   #41
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Quote:
Originally Posted by CRGreathouse View Post
Surely $10^{10000}$ has a specific definition?
At least there is enough vocabulary to express it. Anyway, Richard's paradox comes to mind concerning definability of real numbers.
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May 18th, 2015, 11:43 AM   #42
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In connection with the discussions carried of by the you gentleman, I'd like to point an old article written by Tarski in 1931 (in the collection Logic, Semantics and Metamathematics), whose introduction I reproduce here below:

VI
ON DEFINABLE SETS OF REAL NUMBERS

"MATHEMATICIANS, in general, do not like to deal with the notion of definability; their attitude toward this notion is one of distrust and reserve. The reasons for this aversion are quite clear and understandable. To begin with, the meaning of the term 'definable' is not unambiguous: whether a given notion is definable depends on the deductive system in which it is studied, in particular, on the rules of definition which are adopted and on the terms that are taken as primitive. It is thus possible to use the notion of definability only in a relative sense. This fact has often been neglected in mathematical considerations and has been the source of numerous contradictions, of which the classical example is furnished by the well-known antinomy of Richard. The distrust of mathematicians towards the notion in question is reinforced by the current opinion that this notion is outside the proper limits of mathematics altogether. The problems of making its meaning more precise, of removing the confusions and misunderstandings connected with it, and of establishing its fundamental properties belong to another branch of science -metamathematics.
In this article I shall try to convince the reader that the opinion just mentioned is not altogether correct. Without doubt the notion of definability as usually conceived is of a metamathematical origin. I believe that I have found a general method which allows us to construct a rigorous metamathematical definition of this notion.' Moreover, by analyzing the definition thus obtained it proves to be possible (with some reservations to be discussed at the end of ยง 1) to replace it by a definition formulated exclusively in mathematical terms. Under this new definition the notion of definability does not differ from other mathematical notions and need not arouse either fears or doubts; it can be discussed entirely within the domain of normal mathematical reasoning.
A description of the method in question which is quite general and abstract would involve certain technical difficulties, and, if given at the outset, would lack that clarity which I should like it to have. For this reason I prefer in this article to restrict consideration to a special case, one which is particularly important from the point of view of the questions which interest mathematicians at the present time ; indeed, I shall analyse the notion of definability for just one category of objects, namely, sets of real numbers. Moreover, my considerations will be in the nature of a sketch; I shall content myself simply with constructing some precise definitions, either omitting the consequences which follow from them, or presenting them without demonstration."
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May 18th, 2015, 02:14 PM   #43
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Quote:
Originally Posted by CRGreathouse View Post
The number of microstates in ~2692 oxygen atoms? Oxygen has an entropy of about 205.2 J/(mol*K), so putting the appropriate multiple of Avogadro's number into Boltzmann's entropy formula gives about 2^33 219.1 microstates:
2^(2692 * 1 mol/ (2 * Avogadro's number) * 205.2 J/(mol*K) / (1.38065 * 10^-23 J/K))
Good call.

-Dan
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