Pete Olcott

2018-02-12 16:42:38 UTC

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<a class="moz-txt-link-freetext" href="http://liarparadox.org/index.php/2018/02/12/simple-refutation-of-the-1931-incompleteness-theorem/">http://liarparadox.org/index.php/2018/02/12/simple-refutation-of-the-1931-incompleteness-theorem/</a><br>

<br>

An {a priori fact} is an expression X

of (natural or formal) language L that<br>

has been assigned the semantic property

of True.<br>

<br>

A set T of {a priori facts} of language

L forms the ultimate foundation of the<br>

notion of Truth in language L.<br>

<br>

To verify that an expression X of

language L is True or False only requires<br>

a syntactic logical consequence

inference chain (formal proof) from X or ~X<br>

to one or more elements of T.<br>

<br>

Since an expression X of language L is

not a statement of language L unless<br>

and until it is proven to be True or

False in L, every statement of language<br>

L can be proved or disproved in L.<br>

<br>

<b><span style="background: #ffff00">∀L

∈ Formal_Systems</span></b><b><br>

</b><b>

</b><b><span style="background: #ffff00">∀X

∈ L</span></b><b><br>

</b><b>

</b><b><span style="background: #ffff00">Statement(X)

→ ( Provable(L, X) ∨ Refutable(L, X) )</span></b><b><br>

</b><b>

</b><br>

<b>Stanford Encyclopedia of Philosophy

Gödel’s Incompleteness Theorems</b><b><br>

</b><b>

</b>The first incompleteness theorem states

that in any consistent formal system F<br>

within which a certain amount of

arithmetic can be carried out, <b><span style="background: #ffff00">there

are</span></b><b><br>

</b><b>

</b><b><span style="background: #ffff00">statements

of the language of F which can neither be proved nor disproved

in F.</span></b><br>

<br>

<b>Copyright 2017, 2018 Pete Olcott </b><br>

<br>

<span style="background: #ffff00"></span>

<div class="moz-signature">-- <br>

<meta charset="UTF-8">

<font face="Segoe UI Symbol, sans-serif"><font size="2"><b>

∀X True(X) ↔ ∃Γ ⊆ Axioms Provable(Γ, X) </b></font>

</font></div>

</body>

</html>

<head>

<meta http-equiv="content-type" content="text/html; charset=utf-8">

</head>

<body text="#000000" bgcolor="#FFFFFF">

<a class="moz-txt-link-freetext" href="http://liarparadox.org/index.php/2018/02/12/simple-refutation-of-the-1931-incompleteness-theorem/">http://liarparadox.org/index.php/2018/02/12/simple-refutation-of-the-1931-incompleteness-theorem/</a><br>

<br>

An {a priori fact} is an expression X

of (natural or formal) language L that<br>

has been assigned the semantic property

of True.<br>

<br>

A set T of {a priori facts} of language

L forms the ultimate foundation of the<br>

notion of Truth in language L.<br>

<br>

To verify that an expression X of

language L is True or False only requires<br>

a syntactic logical consequence

inference chain (formal proof) from X or ~X<br>

to one or more elements of T.<br>

<br>

Since an expression X of language L is

not a statement of language L unless<br>

and until it is proven to be True or

False in L, every statement of language<br>

L can be proved or disproved in L.<br>

<br>

<b><span style="background: #ffff00">∀L

∈ Formal_Systems</span></b><b><br>

</b><b>

</b><b><span style="background: #ffff00">∀X

∈ L</span></b><b><br>

</b><b>

</b><b><span style="background: #ffff00">Statement(X)

→ ( Provable(L, X) ∨ Refutable(L, X) )</span></b><b><br>

</b><b>

</b><br>

<b>Stanford Encyclopedia of Philosophy

Gödel’s Incompleteness Theorems</b><b><br>

</b><b>

</b>The first incompleteness theorem states

that in any consistent formal system F<br>

within which a certain amount of

arithmetic can be carried out, <b><span style="background: #ffff00">there

are</span></b><b><br>

</b><b>

</b><b><span style="background: #ffff00">statements

of the language of F which can neither be proved nor disproved

in F.</span></b><br>

<br>

<b>Copyright 2017, 2018 Pete Olcott </b><br>

<br>

<span style="background: #ffff00"></span>

<div class="moz-signature">-- <br>

<meta charset="UTF-8">

<font face="Segoe UI Symbol, sans-serif"><font size="2"><b>

∀X True(X) ↔ ∃Γ ⊆ Axioms Provable(Γ, X) </b></font>

</font></div>

</body>

</html>