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Meta language/addition/algorithm/sum/Gauss/Genz: the sum of the numbers from 1 to 100 is 5050 = 101 x 50:
Example 1 to 10:
1+2+3+4+5+6+7+8+9+10 = (1+10)+(2+9)+(3+8)+(4+7)+(5+6) = 11+11+11+11+11 = 5 x 11 = 55
The sum can be rearranged in such a way that the result of the addition is independent of the sequence of the numbers due to the algorithm.
N.B.: this is a statement about the results of additions, in meta-language.
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Object language.
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Meta language/blackening/characters/formalisms/Hofstadter/Genz: Example for a purely typographical derivation: if 0+0=0, 1+0= 1 etc. as well as 1 = 1 is specified, you can add 1 + x = 1 + x for any x.
Derivation/Formalism/Genz: that negative numbers must be excluded here has no significance for formalism and cannot be used to justify derivations within it.
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Derivation,
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Derivability. >
Formalization.
Hofstadter/Genz: Hofstadter uses the successor relation SS0 instead of 2, so no meanings crept in.
Evidence/Hofstadter: evidence is something informal. The result of reflection.
Formalisation/Hofstadter: formalisation serves to logically defend intuitions.
Derivation/Hofstadter: derivation artificially produced an equivalent of the evidence...
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...that makes the logical structure explicit.
Simplicity/derivation/Hofstadter: it may be that myriads of steps are necessary, but the logical structure turns out to be quite simple.
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Simplicity.
Meaning/Genz: the infinite sequence of the above statements is summed up in the sentence that all numbers, if multiplied by 0, remain unchanged. N.B.: however, this is not based on the meaning of the symbols, but only on the typographic derivation rules of the object language.
Meta language/Genz: it is an insight into formalism that guarantees that all tokens are true.
Object language: be so that the above generalization ("all numbers, multiplied by 0, remain unchanged") can be formulated in it, but cannot be derived.
1st meta-language: here it can be derived. It contains complete induction.
2nd meta-language: here it cannot be derived, but its negation! (see below)
Both meta languages contain the object language. Therefore, the consequences can be derived from them.
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Object language: not all true sentences can be derived in the object language.
Solution: we add the sentence to the language ourselves, then it is true as well as (trivially) derivable.
N.B.: in the second meta-language, which is incompatible with the first, its negation can be added instead of the sentence without creating a contradiction.
2nd meta-language: the 2nd meta-language forces the occurrence of "unnatural" numbers, which cannot be represented as successors of 0.
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1. Douglas Hofstadter (2008). Gödel, Escher, Bach. Stuttgart: Klett-Cotta. p. 240.