12

All mathematical formalizations of (intuitive) computability are known to be equivalent, in particular they are all equivalent to computability on the universal Turing machine. So technological implementation is irrelevant. The Church–Turing thesis states that this coincides in scope with what is "computable by a human being" unconstrained by limitations of ...


11

It is a natural idea, but unfortunately the answer is no, it is not feasible. The root of incompleteness is not numbers, but the possibility of (implicit) self-reference, arithmetic is just the simplest structure that already realizes that possibility. In fact, one does not even need the Peano arithmetic, but a much weaker Robinson arithmetic without even ...


10

The same effect can be achieved with a single sentence:"This sentence is false". It is known as the Liar paradox and goes back to an ancient sophist Epimenides. Your two sentences simply split the Liar in two. There is no endless regress though, it ends in one step. We accept both sentences as "axioms", i.e. "true", but the second sentence implies that the ...


8

He did not write it anywhere. The quote itself only calls Gödel and Skolem "alleged proponents", and later in the article Eklund remarks that "the (supposed) evidence that Skolem adhered to first-order logic is that Skolem held that set theory and arithmetic should be given first-order axiomatizations, whereas... evidence that Gödel adhered to first-order ...


8

If we have an axiom system with a finite number of axioms, we can always reduce them to only one, replacing the set of original axioms with their conjunction. Thus, every non-trivial axiom system that is finitely axiomatized can be formulated in an equivalent form with a single axiom. Gödel's Incompleteness Theorems apply to systems that (in addition to ...


6

The notion is important in mathematical logic and model theory, but not in classical mathematics, including real analysis as traditionally understood. Definable predicates are generally important in the theory of formal systems because they show how expressive they are, for example Tarski's theorem on the undefinability of truth states that in a consistent ...


6

There is a theory of arguments, but I am afraid that the OP conception of argument is too idealized, and the notion of effective debate too narrow, to apply to most of them. If people argued from sets of established axioms and the only issue was whether those sets are equivalent they'd be proving mathematical theorems and meta-theorems of mathematical logic ...


5

Yes, the axioms do trivially prove themselves. Your last derivation, however, is not valid: "A=A" can not be substituted for A because the latter is a symbol in a formal system, while the former is an object of it. You are free to postulate identity law as applied to symbols or laws, of course, in addition to just the identity law for objects, but ...


5

Whether there is a distinction, and what the distinction consists in, is a hotly debated topic. Here are a few things that are typically claimed to be essential to logic: Universal applicability: the laws of logic apply to every subject matter. This would mean that, e.g., different theories of arithmetic have the same underlying logic (usually something ...


4

There is no agreed upon example of this kind. Let us explore the issues: First, we cannot even come up with a decent example of a problem not solvable by any formal system. If you state the problem you want to use, I can simply put solving that problem as a basic entity in a formal system, and thus find a formal system to solve it. If you disagree that ...


4

FOL is the "natural" logic environment to formalize mathematical theories. Propositional calculus, instead, is only a "toy": it is based on a very simplified "model" of language that is not useful to develop interesting theories, but can be used efficiently to study the basic properties of a formal system : consistency, completeness, etc. With FOL we have ...


3

To paraphrase Joel Hamkins answer (pointed out by user4894) on the notion of undefinable reals. The naive account of undefinability points out there is only a countable number of ways we can describe a number, but there is an uncountable number of reals, hence there must be reals that we can't describe; however the notion of definability is problematical: ...


3

Running off the idea that there are undefinable numbers because there are countably infinite definable real numbers and uncountably infinite real numbers, one result that could be considered "important" is that no continuous chaotic system can be perfectly modeled by a Turing machine. Thus, if a truly chaotic dynamic system exists, it could not be part of a ...


3

GIT only applies to axiom systems that can express the natural numbers. Given any such one-axiom system, it would be incomplete. An example of a one-axiom system that's incomplete would be the conjunction of the Godel-Bernays axioms. http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory In other words, NGB can be finitely ...


3

Formal reasoning is an important tool to make subtle differences transparent and explicit. This can be a prerequisite to assess the validity of certain argumentations. E.g., the correct definition of continutiy of a function "f" at an argument "X_0" is ∀epsilon > 0 ∃delta > 0 : |x-x_0| < delta => |f(x) - f(x_0)| < epsilon. On the opposite, the ...


3

I think the confusion comes from schematic identification of logicism with realism, intuitionism with conceptualism, and formalism with nominalism, referencing positions in the old debate on the nature of universals. This is mostly right, but not quite: Hilbert is a nominalist about mathematical objects, but he is a conceptualist (Kantian) about mathematical ...


3

We have to remember that : a ∈ {x:ψ(x)} ⇔ ψ(a) (this is the definition of the "set-builder" symbol { _ : __ } ). But {x:ψ(x)} = Ø, and thus : for all a, a ∉ {x:ψ(x)} ⇔ for all a, ¬ψ(a).


3

You're correct that moving from the integers to the rationals does not fit, because the generalisation that inductive reasoning refers to is a generalisation of statements (or predicates) not of the objects themselves. For example, you could generalise from the statement "all the even numbers above 3 we ever tried can be written as the sum of two primes" ...


3

It is reasonable to mention Wittgenstein in this connection. Brouwer is also relevant as he saw logic as secondary to mathematics. In Brouwers case it is rather clear-cut that he rejected logic as a norm. For him mathematics was an activity in the mind of the mathematician and this determines what can and cannot not be done. The mathematical activity is ...


3

The computable numbers are not technology dependent. A universal computer can simulate any finite physical system to any desired degree of accuracy. And it can simulate not just the input and the output, but also the stages intermediate between the input and output with any desired degree of accuracy: http://www.daviddeutsch.org.uk/wp-content/ItFromQubit....


3

I would do the following: 1. ∃x Cube(x) ∧ Small(d) 2. ∃x Cube(x) ∧Elim1 3. Small(d) ∧Elim1 4. | Cube(z) A 5. | Cube(z) ^ Small(d) ^Intr 3,4 6. | ∃x(Cube(x) ^ Small(d)) ∃Intr 5 7. ∃x(Cube(x) ^ Small(d)) ∃Elim 2,4-6


3

Edit You can only use conjunctive, disjunctive, and negation intro/elim and only uses TautCon for DeMorgans. You also seem to be using rules called "contradiction introduction", and "contradiction elimination". So I suspect what you call "negation elimination" is what is more usually called "double negation elimination". Anyway, you have the first five ...


3

There are two questions here (rephrased): "Where can I find an accessible introduction to Bunge's philosophy?" and "Why is Bunge not more frequently referred to?" Let's start with the first question, which can be answered quickly. A good introduction to Bunge's work, similar to what you would find in an encyclopedia article, is the ...


2

Hilary Putnam argues that formal systems cannot even be used to perform hypothesis selection. Whether you wish to characterize this as "prescription" is up to you, but one might suspect that if formal systems cannot serve here exclusively, they also cannot serve exclusively in other domains with a prescriptive element (e.g. ethics). Epistemic Values are ...


2

For your question I would recommend the preface of Henri Poincaré's Science and Hypothesis (1905), particularly this piece: For [in mathematics] the mind may affirm because it lays down its own laws; but let us clearly understand that while these laws are imposed on our science, which otherwise could not exist, they are not imposed on Nature. Are they ...


2

ZFC is simple enough to axiomatise easily, and be accepted by the majority of mathematicians - some did not - and complex enough to ask good questions that have amenable answers. Once axiomatised and understood sufficiently well, one can look at more complex set theories using ZFC as a base. One angle in more complex set theories is to think of the axiom ...


2

Given that analytical philosophy is closely tied to analysis of how language is used, I would suggest that a close examination of the word concept and how it is used in philosophical discourse (as opposed to its ordinary usage). This does mean rather examining a large corpus of works, and one would need to decide whether one should look at solely analytical ...


2

For the sake of concreteness let's consider the notion of fatherhood (I'm avoiding the use of 'concept' here because we'll be giving it a technical meaning). From experience we have a certain (probably) informal conception of fatherhood. We know, for example, that everyone has a unique father. We know that no one is his own father. We know that two persons ...


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