Principles of bar induction and continuity on Baire space

Kawai, Tatsuji

doi:10.4115/jla.2019.11.ft3

Cited by 2 publications

(4 citation statements)

References 3 publications

(8 reference statements)

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“…We show that

sans-serifFC

is equivalent to the following variant of Bar Induction introduced in :

For any c-bar P \subseteq N^{*} and a subset Q \subseteq N^{*}, if P \subseteq Q and Q is inductive, then Q false(false⟨ false⟩ false) .

Here, a subset

Q \subseteq {double-struckN}^{*}

is inductive if

false(\forall a \in N^{*} false) false[(\forall n \in N) Q (a * false⟨ n false⟩) \Rightarrow Q (a) false]

.…”

Section: Strongly Continuous Functionsmentioning

confidence: 99%

“…It is shown in that, under the assumption of

{AC}_{ω}

, the statement

c - BI

(cf. § 3) is equivalent to the following statement:

Every pointwise continuous function F : N^{N} \to double-struckN is realisable.

This now becomes a corollary of Theorem and Theorem .…”

Section: Brouwer‐operationsmentioning

confidence: 99%

“…We are ready to state the main result of this section. It is shown in [10] that, under the assumption of AC ω , the statement c-BI (cf. § 3) is equivalent to the following statement:…”

Section: Casementioning

confidence: 99%

“…We are ready to state the main result of this section. It is shown in [10] that, under the assumption of AC ω , the statement c-BI (see Section 3) is equivalent to the following statement: UC B Every pointwise continuous function F : N N → N is realisable. This now becomes a corollary of Theorem 3.9 and Theorem 4.11.…”

mentioning

confidence: 99%

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Formally continuous functions on Baire space

Kawai

2018

Mathematical Logic Qtrly

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A function from Baire space NN to the natural numbers double-struckN is called formally continuous if it is induced by a morphism between the corresponding formal spaces. We compare formal continuity to two other notions of continuity on Baire space working in Bishop constructive mathematics: one is a function induced by a Brouwer‐operation (i.e., inductively defined neighbourhood function); the other is a function uniformly continuous near every compact image. We show that formal continuity is equivalent to the former while it is strictly stronger than the latter. The equivalence of formally continuous functions and those induced by Brouwer‐operations requires Countable Choice.

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“…We show that

sans-serifFC

is equivalent to the following variant of Bar Induction introduced in :

For any c-bar P \subseteq N^{*} and a subset Q \subseteq N^{*}, if P \subseteq Q and Q is inductive, then Q false(false⟨ false⟩ false) .

Here, a subset

Q \subseteq {double-struckN}^{*}

is inductive if

false(\forall a \in N^{*} false) false[(\forall n \in N) Q (a * false⟨ n false⟩) \Rightarrow Q (a) false]

.…”

Section: Strongly Continuous Functionsmentioning

confidence: 99%

“…It is shown in that, under the assumption of

{AC}_{ω}

, the statement

c - BI

(cf. § 3) is equivalent to the following statement:

Every pointwise continuous function F : N^{N} \to double-struckN is realisable.

This now becomes a corollary of Theorem and Theorem .…”

Section: Brouwer‐operationsmentioning

confidence: 99%

“…We are ready to state the main result of this section. It is shown in [10] that, under the assumption of AC ω , the statement c-BI (cf. § 3) is equivalent to the following statement:…”

Section: Casementioning

confidence: 99%

mentioning

confidence: 99%

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