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Gödel‘s Second Incompleteness Theorem –
this theorem states the following: “For any
formal effectively generated theory
T
including basic arithmetical truths and also certain
truths about formal provability,
T
includes a statement of its own consistency if and only
if
T
is inconsistent.” (1 wikipedia)
Mimicking examples of this claim are found in, “theories of real numbers, of complex
numbers and of Euclidean geometry.”(2 http://www.math.hawaii.edu/) They do have
absolute complete axiomatizations. Hence in these systems there is no such thing as true
but unprovable sentences. But this escapes the conclusion of first incompleteness
theorem! The reason for this is the system’s inadequacy or in other words its
‘inconsistent’ nature.
The above points to the fact that Gödel‘s second theorem strengthens his first theorem. If
we examine the first incompleteness theorem it does not “directly express the consistency
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This note was uploaded on 01/27/2011 for the course PHI 330 taught by Professor Mar during the Fall '10 term at SUNY Stony Brook.
 Fall '10
 Mar

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