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Damjan Bojadiev
Reflexive extension of reflexive systems II
(CogFest 1996, LJ An? liza, 1999)
nothing (new) to it
? basic idea doesnt (seem to) work
? trivial case of known results/definitions
S ? Scons(S) ? Scons(S)cons(Scons(S)) ?
Scons(Scons(S))cons(Scon
CogSci relevance Lucas-Penrose argument
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Reflexive systems
Non-reflexive extensions

S
S
S
3
Reflexive systems
Non-reflexive extensions

4
Reflexive systems
Non-reflexive extensions

X
X is blind to the effect of
its own addition
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Reflexive systems
reflexive extensions

X
6
Reflexive systems
reflexive extensions

X
7
Reflexive systems
reflexive extensions

X
X
X incorporates the effect of
its own addition
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Possible (recursion theorem, Kleene)
but inconsistent (Gödel)
cons(Sself)
S
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cons(Sself) ?? P
P
S
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cons(Sself) ?? cons(Scons(S))
cons(Scons(S))
S
cons(Sself) ?? cons(S)
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But to formulate cons(Scons(S))
cons(Scons(S))
S
we have to formulate Scons(S)
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But to formulate cons(Scons(S))
cons(Scons(S))
cons(S)
S
we have to formulate Scons(S)
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S ? Scons(S) ? Scons(S)cons(Scons(S)) ?
Scons(Scons(S))cons(Scon
Lucas-Penrose argument not just how far, but
also how fast
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S? ? S?cons(S?) ? S?cons(S?)cons(S?cons(S?))
? S?cons(S?)cons(S?
S ? Scons(S) ? Scons(S)cons(Scons(S)) ?
Scons(Scons(S))cons(Scon
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ax(x)
ax(x) ? xcons(ax(x))

S? ? S?cons(S?) ? S?cons(S?)cons(S?cons(S?))
? S?cons(S?)cons(S?
S ? Scons(S) ? Scons(S)cons(Scons(S)) ?
Scons(Scons(S))cons(Scon
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ax(x)
ax(x) ? xcons(ax(x))

Ordinal n !
Any limit ordinal
S? ? S?cons(S?) ? S?cons(S?)cons(S?cons(S?))
? S?cons(S?)cons(S?
S ? Scons(S) ? Scons(S)cons(Scons(S)) ?
Scons(Scons(S))cons(Scon
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?0
ax(x)
?1
ax(x) ? xcons(ax(x))

??
Ordinal n
Any limit ordinal
?lim(e) (?n) ?e(n)
?0, ?1, ?2,
?0, ?2, ?4,
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Franzen, Torkel, Inexhaustibility A
Non-Exhaustive Treatment, Lecture Notes in Logic
16. A. K. Peters 2004.
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Franzen, Torkel, Inexhaustibility A
Non-Exhaustive Treatment, Lecture Notes in Logic
16. A. K. Peters 2004.
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