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| {{For|the concept of objects in philosophy|Object (philosophy)}}
| | Among the drugs that doctors recommend are: acyclovir, valacyclovir and famciclovir. Shifting forward, herbal treatment for herpes can also assist and you might opt for on distinctive variety of possibilities for treatment method as very well. In fact herpes as with many sexually transmitted diseases has been rising every year and with people's behavior there doesn't seem that there is much we can do other then educate. Infected fish that survive the outbreak could also become carriers. Replication of the herpes simplex virus takes place in the nucleus of the host cell. <br><br>* Genital herpes is curable by taking antiviral drugs like Acyclovir. 38-41 (2005), "Use of complementary and alternative medicine for the treatment of genital herpes. Ice will still provide great comfort, and reduce swelling. These blisters break leaving sores that may take several weeks to heal. Genital acne is easily confused with particular sexually transmitted infections due to similarity in symptoms, and, is "not" alike to ones on the face; but treatment is pretty much the same. <br><br>No, prescribing medication and performing surgery is the lifeblood of their business. If the man has oral herpes, he should avoid performing oral sex while cold sores are present. Can monolaurin, a nutrient from coconut oil, lauric acid (that you can buy online or in a health food store) also dissolve the swine flu virus. In the past, you had to have a herpes symptom present to make a diagnosis, but now there are blood antibody tests that detect herpes infection even if you've never had a symptom. Hopefully, this report was ready to remedy the question of all those asking what does genital herpes appear like. <br><br>Perhaps these latest developments will bring about the breakthroughs that are so desperately needed. Since women frequently experience their herpes sores in the vagina, many have mistaken the few herpes symptoms that they have are cause by some other type of infection. You can pass genital herpes to someone else even when you experience no symptoms. If a woman already has genital herpes and knows it, the risk of her giving it to her baby is very small indeed. Genital herpes symptoms in men can be managed through diet, medication or natural treatments. <br><br>This was to be done a few times a day throughout the outbreak, and for the next several times an outbreak was surfacing. Lesions typically last for 9 days, and viral shedding lasts for approximately 4 days. Or if a mild break out takes place and then clears up quickly you might believe that the bumps or soreness couldn't be genital herpes given that it cleared up by itself. A breakout or flare-up of genital herpes looks like big blisters or pus filled sores and they will appear on a guy or a female who lugs the herpes virus. Mc - Donnell Foundation and the National Institutes of Health.<br><br>If you have any queries with regards to where by and how to use herpes diet ([http://www.sohen.info/ http://www.sohen.info]), you can call us at our internet site. |
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| '''Object theory''' is a theory in [[philosophy]] and [[mathematical logic]] concerning objects and the statements that can be made about objects.{{citation needed|date=July 2012}}
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| In some cases "objects" can be concretely thought of as symbols and strings of symbols, here illustrated by a string of four symbols " ←←↑↓←→←↓" as composed from the 4-symbol alphabet { ←, ↑, →, ↓ } . When they are "known only through the relationships of the system [in which they appear], the system is [said to be] ''abstract'' ... what the objects are, in any respect other than how they fit into the structure, is left unspecified." (Kleene 1952:25) A further specification of the objects results in a '''model''' or '''representation''' of the abstract system, "i.e. a system of objects which satisfy the relationships of the abstract system and have some further status as well" (ibid).
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| A system, in its general sense, is a collection of '''objects''' O = {o<sub>1</sub>, o<sub>2</sub>, ... o<sub>n</sub>, ... } and (a specification of) the '''relationship''' ''r'' or relationships r<sub>1</sub>, r<sub>2</sub>, ... r<sub>n</sub> between the objects:
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| : Example: Given a simple system = { { ←, ↑, →, ↓ }, '''∫''' } for a very simple relationship between the objects as signified by the symbol '''∫''' :<ref>Abstractly, the relationship '''∫''' is defined by the collection of ordered pairs { ( →, ↑ ), ( ↑, ← ), ( ←, ↓ ), (↓, →) }</ref>
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| :: '''∫'''→ => ↑, '''∫'''↑ => ←, '''∫'''← => ↓, '''∫'''↓ => →
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| A model of this system would occur when we assign, for example the familiar natural numbers { 0, 1, 2, 3 }, to the symbols { ←, ↑, →, ↓ }, i.e. in this manner: → = 0, ↑ = 1, ← = 2, ↓ = 3 . Here, the symbol '''∫''' indicates the "successor function" (often written as an apostrophe ' to distinguish it from +) operating on a collection of only 4 objects, thus 0' = 1, 1' = 2, 2' = 3, 3' = 0.
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| :Or, we might specify that '''∫''' represents 90-degree counter-clockwise rotations of a simple object → .
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| == The genetic versus axiomatic method ==
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| The following is an example of the '''genetic''' or '''constructive''' method of making objects in a system, the other being the '''axiomatic''' or '''postulational''' method. Kleene states that a genetic method is intended to "generate" all the objects of the system and thereby "determine the abstract structure of the system completely" and uniquely (and thus define the system '''categorically'''). If axioms rather than a genetic method is used, such axiom-sets are said to be '''categorical'''.<ref>Kleene 1952:26. This distinction between the constructive and axiomatic methods, and the words used to describe them, are Kleene's per his reference to Hilbert 1900.</ref>
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| Unlike the '''∫''' example above, the following creates an unbounded number of objects. The fact that O is a set, and □ is an element of O, and ■ is an operation, must be specified at the outset; this is being done in the language of the [[metatheory]] (see below):
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| : Given the system ( O, □, ■ ): O = { □, ■□, ■■□, ■■■□, ■■■■□, ■■■■■□, ..., ■<sup>n</sup>□, etc. }
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| == Abbreviations ==
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| The object ■<sup>n</sup>□ demonstrates the use of "abbreviation", a way to simplify the denoting of objects, and consequently discussions about them, once they have been created "officially". Done correctly the definition would proceed as follows:
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| ::: ■□ ≡ ■<sup>1</sup>□, ■■□ ≡ ■<sup>2</sup>□, ■■■□ ≡ ■<sup>3</sup>□, etc, where the notions of ≡ ("defined as") and "number" are presupposed to be understood intuitively in the metatheory.
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| Kurt Gödel 1931 virtually constructed the entire proof of his [[incompleteness theorem]]s (actually he proved Theorem IV and sketched a proof of Theorem XI) by use of this tactic, proceeding from his axioms using substitution, concatenation and deduction of ''modus ponens'' to produce a collection of 45 "definitions" (derivations or theorems more accurately) from the axioms.
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| A more familiar tactic is perhaps the design of subroutines that are given names, e.g. in Excel the subroutine " =INT(A1)" that returns to the cell where it is typed (e.g. cell B1) the integer it finds in cell A1.
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| ==Models==
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| A '''model''' of the above example is a left-ended [[Post-Turing machine]] tape with its fixed "head" located on the left-end square; the system's relation is equivalent to: "To the left end, tack on a new square □, right-shift the tape, then print ■ on the new square". Another model is the natural numbers as created by the "successor" function. Because the objects in the two systems e.g. ( □, ■□, ■■□, ■■■□ ... ) and (0, 0', 0'', 0''', ...) can be put into a 1-1 correspondence, the systems are said to be (simply) '''[[isomorphic]]''' (meaning "same shape"). Yet another isomorphic model is the little sequence of instructions for a [[counter machine]] e.g. "Do the following in sequence: (1) Dig a hole. (2) Into the hole, throw a pebble. (3) Go to step 2."
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| As long as their objects can be placed in one-to-one correspondence ("while preserving the relationships") models can be considered "equivalent" no matter how their objects are generated (e.g. genetically or axiomatically):
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| :"Any two simply isomorphic systems constitute representations [models] of the same abstract system, which is obtained by abstracting from either of them, i.e. by leaving out of account all relationships and properties except the ones to be considered for the abstract system." (Kleene 1935:25)
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| == Tacit assumptions, tacit knowledge ==
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| An alert reader may have noticed that writing symbols □, ■□, ■■□, ■■■□, etc. by concatenating a marked square, i.e. ■, to an existing string is different than writing the completed symbols one after another on a Turing-machine tape. Another entirely possible scenario would be to generate the symbol-strings one after another on different sections of tape e.g. after three symbols: ■■■□■■□■□□. The proof that these two possibilities are different is easy: they require different "programs". But in a sense both versions create the same objects; in the second case the objects are preserved on the tape. In the same way, if a person were to write 0, then erase it, write 1 in the same place, then erase it, write 2, erase it, ad infinitum, the person is generating the same objects as if they were writing down 0 1 2 3 ... writing one symbol after another to the right on the paper.
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| Once the step has been taken to write down the symbols 3 2 1 0 one after another on a piece of paper (writing the new symbol on the left this time), or writing ∫∫∫※∫∫※∫※※ in a similar manner, then putting them in 1-1 correspondence with the Turing-tape symbols seems obvious. Digging holes one after the other, starting with a hole at "the origin", then a hole to its left with one pebble in it, then a hole to ''its'' left with two pebbles in it, ad infinitum, raises practical questions, but in the abstract it too can be seen to be conducive to the same 1-1 correspondence.
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| However, nothing in particular in the definition of genetic versus axiomatic methods clears this up—these are issues to be discussed in the metatheory. The mathematician or scientist is to be held responsible for sloppy specifications. Breger cautions that axiomatic methods are susceptible to tacit knowledge, in particular, the sort that involves "know-how of a human being" (Breger 2000:227).
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| == A formal system ==
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| In general, in mathematics a [[formal system]] or "formal theory" consists of "objects" in a structure:
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| * The symbols to be concatenated (adjoined),
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| * The formation-rules (completely specified, i.e. formal rules of [[syntax]]) that dictate how the symbols and the assemblies of symbols are to be formed into assemblies (e.g. sequences) of symbols (called terms, formulas, sentences, propositions, theorems, etc.) so that they are in "well-formed" patterns (e.g. can a symbol be concatenated at its left end only, at its right end only, or both ends simultaneously? Can a collection of symbols be substituted for (put in place of) one or more symbols that may appear anywhere in the target symbol-string?),
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| * Well-formed "propositions" (called "theorems" or assertions or sentences) assembled per the formation rules,
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| * A few [[axiom]]s that are stated up front and may include "undefinable notions" (examples: "set", "element", "belonging" in set theory; "0" and " ' " (successor) in number theory),
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| * At least one rule of [[deductive inference]] (e.g. [[modus ponens]]) that allow one to pass from one or more of the axioms and/or propositions to another proposition.
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| == Informal theory, object theory, and metatheory ==
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| A [[metatheory]] exists outside the formalized object theory—the meaningless symbols and relations and (well-formed-) strings of symbols. The metatheory comments on (describes, interprets, illustrates) these meaningless objects using "intuitive" notions and "ordinary language". Like the object theory, the metatheory should be disciplined, perhaps even quasi-formal itself, but in general the interpretations of objects and rules are intuitive rather than formal. Kleene requires that the methods of a metatheory (at least for the purposes of [[metamathematics]]) be finite, conceivable, and performable; these methods cannot appeal to the [[completed infinite]]. "Proofs of existence shall give, at least implicitly, a method for constructing the object which is being proved to exist."<ref>This is an [[intuitionist]] requirement: It formally proscribes the use of the [[law of excluded middle]] over infinite collections (sets) of objects."</ref> (p. 64)
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| Kleene summarizes this as follows: "In the full picture there will be three separate and distinct "theories""
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| :"(a) the informal theory of which the formal system constitutes a formalization
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| :"(b) the formal system or '''object theory''', and
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| :"(c) the metatheory, in which the formal system is described and studied" (p. 65)
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| He goes on to say that object theory (b) is not a "theory" in the conventional sense, but rather is "a system of symbols and of objects built from symbols (described from (c))". <!-- This object theory (b) he calls (perhaps confusingly) a "model" of the informal theory (a).-->
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| == Expansion of the notion of formal system ==
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| === Well-formed objects ===
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| If a collection of objects (symbols and symbol-sequences) is to be considered "well-formed", an algorithm must exist to determine, by halting with a "yes" or "no" answer, whether or not the object is well-formed (in mathematics a '''wff''' abbreviates [[well-formed formula]]). This algorithm, in the extreme, might require (or be) a [[Turing machine]] or [[Turing-equivalent]] machine that "[[parse]]s" the symbol-string as presented as "data" on its tape; before a [[universal Turing machine]] can execute an instruction on its tape, it must parse the symbols to determine the exact nature of the instruction and/or datum encoded there. In simpler cases a [[finite state machine]] or a [[pushdown automaton]] can do the job. Enderton describes the use of "trees" to determine whether or not a logic formula (in particular a string of symbols with parentheses) is well formed.<ref>Enderton 2002:30</ref> [[Alonzo Church]] 1934<ref>Church 1934 reprinted in Davis 1965:88ff</ref> describes the construction of "formulas" (again: sequences of symbols) as written in his λ-calculus by use of a [[Recursion|recursive]] description of how to start a formula and then build on the starting-symbol using concatenation and substitution.
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| Example: Church specified his λ-calculus as follows (the following is simplified version leaving out notions of free- and bound-variable). This example shows how an object theory begins with a specification of an ''object system'' of symbols and relations (in particular by use of concatenation of symbols):
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| :(1) Declare the symbols: '''{''', '''}''', '''(''', ''')''', '''λ''', '''[''', ''']''' plus an infinite number of ''variables'' '''a''', '''b''', '''c''', ..., '''x''', ...
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|
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| :(2) Define ''formula'': a sequence of symbols
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| :(3) Define the notion of "well-formed formula" (wff) recursively starting with the "basis" (3.i):
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| :*(3.1) (basis) A variable '''x''' is a wff
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| :*(3.2) If '''F''' and '''X''' are wffs, then '''{F}(X)''' is a wff; if '''x''' occurs in '''F''' or '''X''' then it is said to be a variable in '''{F}(X)'''.
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| :*(3.3) If '''M''' is well-formed and '''x''' occurs in '''M''' then '''λx[M]''' is a wff.
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| :(4) Define various abbreviations:
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| :* '''{F}[X]''' abbreviates to '''F(X)''' if '''F''' is a single symbol
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| :* '''<math>{{F}[X]}[Y]</math>''' abbreviates to '''{F}(X,Y)''' or '''F(X,Y)''' if '''F''' is a single symbol
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| :* '''λx<sub>1</sub>λx<sub>2</sub>[...λx<sub>n</sub>[M]...]''' abbreviates to '''λx<sub>1</sub>x<sub>2</sub>...x<sub>n</sub>•M'''
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| :* '''λab•a(b)''' abbreviates to '''1'''
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| :* '''λab•a(a(b))''' abbreviates to '''2''', etc.
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| :(5) Define the notion of "substitution" of formula '''N''' for variable '''x''' throughout '''M'''<ref>The substitution gets complicated and requires more information (e.g. definitions of "free-" and "bound-" variables and three varieties of substitution) than has been given in this brief example.</ref> (Church 1936)
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| <!-- :(6) If the system is going to start with some undefined symbols and some axioms and then create (make) objects in a (logically) deductive manner, the system must specify some relations that are [amount to] logical deduction (e.g. [[modus ponens]]). The "schemata" [??] is a Transformation rules e.g. the rules of logical deduction: Define binary and ternary relations that such as "immediate consequence of" (Godel 1931, Kleene 1952), or "conversion" (Church 1934) with respect to two or three objects a and b and c e.g. the binary operation (a, b) and the [[ternary operation]] ( (a, b), c) where (a, b) are an ordered pair of objects and c is the outcome:
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| Example: [[Kurt Gödel]] 1931<ref>Gödel 1931 reprinted in van Heijenoort 1976:601.</ref> specified the following two forms of "immediate consequence of" with respect to his "objects" that he called "formulas".
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| ::In the following the symbol ≡ signifies "is defined as", ~ signifies the logical NOT, V signifies the logical [[inclusive-OR]] and → signifies "IF ... THEN ..." (logical implication):
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| : (1) Given a ≡ ( b → c ) ≡ ( ~(b) V c ) then by [[modus ponens]] i.e. b & ( b → c ) → c
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| : then b & a → c is a tautology -- true in all circumstances
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| : (2) Given ''a'' is a formula with ''v'' a variable, then ''c'' is an immediate consequence when any value for v plugged into ''a(v)'', i.e. (∀v: a(v)) ≡ c -->
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| === Undefined (primitive) objects ===
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| Certain objects may be "undefined" or "primitive" and receive definition (in the terms of their behaviors) by the introduction of the [[axioms]].
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| In the next example, the undefined symbols will be { ※, '''ↀ''', '''∫''' }. The axioms will describe their ''behaviors''.
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| === Axioms ===
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| Kleene observes that the axioms are made up of two sets of symbols: (i) the undefined or primitive objects and those that are previously known. In the following example, it is previously known in the following system ( O, ※, '''ↀ''', '''∫''' ) that O constitutes a set of objects (the "domain"), ※ is an object in the domain, '''ↀ''' and '''∫''' are symbols for relations between the objects, => indicates the "IF THEN" logical operator, ε is the symbol that indicates "is an element of the set O", and "n" will be used to indicate an arbitrary element of set-of-objects O.
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| After (i) a definition of "string '''S'''"—an object that is a symbol ※ or concatenated symbols ※, ↀ or ∫, and (ii) a definition of "well-formed" strings -- (basis) ※ and ↀ'''S''', ∫'''S''' where '''S''' is any string, come the axioms:
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| * ↀ※ => ※, in words: "IF ↀ is applied to object ※ THEN object ※ results."
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| * ∫n ε O, in words "IF ∫ is applied to arbitrary object "n" in O THEN this object ∫n is an element of O".
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| * ↀn ε O, "IF ↀ is applied to arbitrary object "n" in O THEN this object ↀn is an element of O".
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| * ↀ∫n => n, "IF ↀ is applied to object ∫n THEN object n results."
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| * ∫ↀn => n, "IF ∫ is applied to object ↀn THEN object ※ results."
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| So what might be '''the (intended) interpretation'''<ref>Kleene defines '''the intended interpretation''' as "the meanings which are intended to be attached to the symbols, formulas, etc. of a given formal system, in consideration of the system as a formalization of an informal theory....(p. 64)</ref> of these symbols, definitions, and axioms?
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| If we define ※ as "0", ∫ as "successor", and ↀ as "predecessor" then ↀ※ => ※ indicates "proper subtraction" (sometimes designated by the symbol ∸, where "predecessor" subtracts a unit from a number, thus 0 ∸1 = 0). The string " ↀ∫n => n " indicates that if first the successor is applied to an arbitrary object n and then the predecessor ↀ is applied to ∫n, the original n results." | |
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| Is this set of axioms "adequate"? The proper answer would be a question: "Adequate to describe what, in particular?" "The axioms determine to which systems, defined from outside the theory, the theory applies." (Kleene 1952:27). In other words, the axioms may be sufficient for one system but not for another.
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| In fact, it is easy to see that this axiom set is not a very good one—in fact, it is [[inconsistent]] (that is, it yields inconsistent outcomes, no matter what its interpretation):
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| : Example: Define ※ as 0, ∫※ as 1, and ↀ1 = 0. From the first axiom, ↀ※ = 0, so ∫ↀ※ = ∫0 = 1. But the last axiom specifies that for any arbitrary n including ※ = 0, ∫ↀn => n, so this axiom stipulates that ∫ↀ0 => 0, not 1.
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| Observe also that the axiom set does not specify that ∫n ≠ n. Or, excepting the case n = ※, ↀn ≠ n. If we were to include these two axioms we would need to describe the intuitive notions "equals" symbolized by = and not-equals symbolized by ≠.
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| == See also ==
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| *[[Metatheory]]
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| *[[Object language]]
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| == Footnotes ==
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| <references/>
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| == References ==
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| *Herbert Breger 2000, ''Tacit Knowledge and Mathematical Progress'', in E. Groshoz and H. Breger (eds.) 2000, ''The Growth of Mathematical Knowledge, 221-230. Kluwer Academic Publishers. Dordrecht, Netherlands. ISBN 0-7923-6151-2
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| *[[Alonzo Church]] 1936 ''An Unsolvable Problem of Elementary Number Theory'', reprinted in [[Martin Davis]] 1965, ''The Undecidable'', Raven Press, NY. No ISBN.
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| *Herbert B. Enderton 2001, ''A Mathematical Introduction to Logic: Second Edition'', Harcort Academic Press, Burlington MA. ISBN 978-0-12-238452-3.
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| *[[Stephen C. Kleene]] 1952, 6th reprint 1971, 10th impression 1991, ''Introduction to Metamathematics'', North-Holland Publishing Company, Amsterdam NY, ISBN 0-7204-2103-9.
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| ==External links==
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| *[http://mally.stanford.edu/theory.html The Theory of Abstract Objects] at the Stanford Metaphysical Research Lab.
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| {{DEFAULTSORT:Object Theory}}
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| [[Category:Metalogic]]
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| [[Category:Theories of deduction]]
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