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Results tagged with constructive-mathematics
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user 5442
Constructive mathematics in the style of Bishop, including its semantics using realizabilty or topological methods.
42
votes
Accepted
How Would an Intuitionist Prove This?
In general, when working in constructive mathematics, the strategy for proving $Q \lor R$ is to prove $Q$ or to prove $R$. In this case, just knowing abstractly that "there is an $n \not = 1$ that div …
- 9,683
19
votes
Are all functions in Bishop's constructive mathematics continuous?
Bishop's mathematics is compatible with classical mathematics. For example, if we look at set theory in Bishop's framework, each model V of ZFC is a model of Bishop's system, and if we look at second- …
- 9,683
3
votes
What happens when we print the digits of a real number?
This is slightly tangential but: the problem you are running into is the reason why people in constructive settings use "quickly convergent Cauchy sequences" instead of ordinary Cauchy sequences.
A …
- 9,683
11
votes
Are real numbers countable in constructive mathematics?
You are using the word "constructive" in an unusual way. It is true that, in ZFC, the set of computable real numbers is countable, but that is not directly a statement about constructive mathematics. …
- 9,683
13
votes
Difference between constructive Dedekind and Cauchy reals in computation
Once we look at computational content, rather than constructive content, things are easier to answer. When we work on the level of individual reals, all the representations are equivalent - a real is …
- 9,683
10
votes
Accepted
Did Bishop, Heyting or Brouwer take partial functions seriously?
I don't believe that Bishop explicitly assumed all functions are continuous. I think that "no discontinuous function can be proved to be total in Bishop's constructive mathematics" is actually a very …
- 9,683
4
votes
Cauchy real numbers with and without modulus
Let $(a_n)$ be a Specker sequence - a computable, bounded, increasing sequence of rationals so that the limit is not a computable real.
First, assume for simplicity that we work in any system of s …
- 9,683
10
votes
Accepted
Strength of Bishop style constructive mathematics vs $\mathsf{RCA}_0$
BISH famously includes the full axiom of choice scheme (in the functional language of second-order arithmetic), which is utterly weak in that context but very strong when combined with the law of the …
- 9,683