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Difference between revisions of "Formalization method"

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A way of expressing by a [[Formal system|formal system]] a mathematical theory. It is one of the main methods in [[Proof theory|proof theory]].
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A way of expressing by a [[formal system]] a mathematical theory. It is one of the main methods in [[proof theory]].
  
 
An application of the formalization method involves carrying out the following stages.
 
An application of the formalization method involves carrying out the following stages.
  
1) Putting the original mathematical theory into symbols. In this all the propositions of the theory are written in a suitable logico-mathematical language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040920/f0409201.png" />.
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1) Putting the original mathematical theory into symbols. In this all the propositions of the theory are written in a suitable logico-mathematical language $\mathcal L$.
  
 
2) The deductive analysis of the theory and the choice of axioms, that is, of a collection of propositions of the theory from which all other propositions of the theory can be logically derived.
 
2) The deductive analysis of the theory and the choice of axioms, that is, of a collection of propositions of the theory from which all other propositions of the theory can be logically derived.
  
3) Adding the axioms in their symbolic notation to a suitable [[Logical calculus|logical calculus]] based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040920/f0409202.png" />.
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3) Adding the axioms in their symbolic notation to a suitable [[logical calculus]] based on $\mathcal L$.
  
The system obtained by this formalization is now itself the object of precise mathematical study (see [[Axiomatic method|Axiomatic method]]; [[Proof theory|Proof theory]]).
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The system obtained by this formalization is now itself the object of precise mathematical study (see [[Axiomatic method]]; [[Proof theory]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1951)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1951)</TD></TR>
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</table>
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Latest revision as of 16:48, 18 October 2016

A way of expressing by a formal system a mathematical theory. It is one of the main methods in proof theory.

An application of the formalization method involves carrying out the following stages.

1) Putting the original mathematical theory into symbols. In this all the propositions of the theory are written in a suitable logico-mathematical language $\mathcal L$.

2) The deductive analysis of the theory and the choice of axioms, that is, of a collection of propositions of the theory from which all other propositions of the theory can be logically derived.

3) Adding the axioms in their symbolic notation to a suitable logical calculus based on $\mathcal L$.

The system obtained by this formalization is now itself the object of precise mathematical study (see Axiomatic method; Proof theory).

References

[1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)
How to Cite This Entry:
Formalization method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Formalization_method&oldid=17646
This article was adapted from an original article by S.N. Artemov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article