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But any expression can always be written as a string using something like Mathematica FullForm .
And if one is going to reproduce all the equivalences that hold for a particular form then these constraints must in effect be such as to force that form to occur. … Every axiom system must allow an operator of at least some form. … They yield respectively junctional, equivalential, implicational and full propositional or sentential calculus (ordinary logic).
After just 29 steps, this edge takes on a form that repeats every 1700 steps. During each such cycle, a total of 65 persistent structures are produced, of 11 of the 15 kinds from page 292 , and their interactions make the full repetition period 6800 steps.
For in the late 1950s a whole hierarchy of systems with so-called intermediate degrees were constructed with the property that questions about the ultimate output from their evolution could not in general be answered by finite computation, but for which the actual form of this output was not flexible enough to be able to emulate a full range of other systems, and thus support universality. … And so we might wonder whether perhaps some other form of regularity could be present that would prevent systems like rule 30 from being universal.
It is convenient to think of expressions in a language as having forms such as s["(", "(", ")", ")"] with Attributes[s] = Flat . … (Note that in practice the output from a parser for a context-free language is usually represented as a tree—as in Mathematica FullForm —with each node corresponding to one rule application.)
The theory of types used in Principia Mathematica introduced some distinction, and following the proof of Gödel's Completeness Theorem for first-order logic in 1930 (see page 1152 ) standard axiom systems for mathematics (as given on pages 773 and 774 ) began to be reformulated in first-order form, with set theory taking over many of the roles of second-order logic. In current mathematics, second-order logic is sometimes used at the level of notation, but almost never in its full form beyond.
Indeed, the criss-crossing of veins in the leaves of higher plants may be not unrelated to the fact that stems in the pictures two pages ago often cross over—although certainly many of the veins in actual full-grown leaves are probably added long after the shapes of the leaves are determined. … But I strongly suspect that in fact the very same simple process of branching is ultimately responsible both for the overall forms of plants, and for the shapes of their leaves.
The idea that even a vacuum without particles will have a complicated and in some ways random form also exists in standard quantum field theory in traditional physics. The full mathematical structure of quantum field theory is far from completely worked out.
For then one can start with an expression, convert it to standard form, then convert back to any expression that is equivalent. … A standard form in terms of Nand can be constructed essentially by direct translation of DNF; other methods can be used for the various other operators shown. … (Note that if an axiom system does manage to reproduce logic in full then as indicated on page 814 its consequences can always be derived by proofs of limited length, if nothing else by using truth tables.)
I have tried in the index to give all names in the form they might be used on standardized documents in the modern U.S. … I give in full those forenames that I believe are or were most commonly used by a particular individual; for other forenames (including for example Russian patronymics) I give only initials. I normally give formal versions of forenames—though for individuals I have personally known I give in the text the form of forenames I would normally use in addressing them.