Nuprl Lemma : gen-continuity-contradicts-kuroda
(∀P:(ℕ ⟶ ℕ) ⟶ ℙ. ∀f:ℕ ⟶ ℕ. ((P f) ⇒ ⇃(∃k:ℕ. ∀g:ℕ ⟶ ℕ. ((f = g ∈ (ℕk ⟶ ℕ)) ⇒ (P g)))))
⇒ (¬(∀A:ℕ ⟶ ℙ. ((∀m:ℕ. (¬¬(A m))) ⇒ (¬¬(∀m:ℕ. (A m))))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
is-absolutely-free: is-absolutely-free{i:l}(f)
Lemmas referenced :
all_wf,
nat_wf,
not_wf,
quotient_wf,
exists_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
true_wf,
equiv_rel_true,
ge_wf,
zero-seq_wf,
Kripke2a,
increasing-zero-seq,
Kripke2b,
init0-zero-seq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
because_Cache,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
instantiate,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
cumulativity,
universeEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
intEquality
Latex:
(\mforall{}P:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((P f) {}\mRightarrow{} \00D9(\mexists{}k:\mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((f = g) {}\mRightarrow{} (P g)))))
{}\mRightarrow{} (\mneg{}(\mforall{}A:\mBbbN{} {}\mrightarrow{} \mBbbP{}. ((\mforall{}m:\mBbbN{}. (\mneg{}\mneg{}(A m))) {}\mRightarrow{} (\mneg{}\mneg{}(\mforall{}m:\mBbbN{}. (A m))))))
Date html generated:
2017_09_29-PM-06_09_52
Last ObjectModification:
2017_07_26-PM-03_12_43
Theory : continuity
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