16

If you make the supposition that no thing inside the universe could generate the universe and that every thing that exists is inside the universe than the direct conclusion is that the universe was not generated by a thing, which is similar to saying that it was generated by nothing. The difficulty here is that you have to ask yourself what you mean by "...


13

In ZFC we have two axioms that settle that question: Empty Set. There is a set that contains nothing. Extensionality. If sets A and B have exactly the same members, then A = B. The Empty Set Axiom allows us to conclude that there is an empty set. Suppose there are two empty sets A and B. Vacuously, every member of A is a member of B (since A has no ...


12

Russell's paradox arises within naïve set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself. Hence the paradox. The "root" of the paradox is the so-called unrestriceted Comprehension Principle of naïve set theory: for every property φ(x) ...


9

The the set {¬,∨,∧,⇒,⇔} is usually used because it includes the "most natural" ones. If we start with a minimal set, like {¬,⇒}, the other ones are usually introduced as abbreviations. There is no "deep" reason : mainly tradition, and a "reasonable" trade-off between savings (minimality) and readibility (to express p ∧ q as ¬(p ⇒ ¬q) is not so "natural".


7

Logic, paraconsistent or not, does not exactly make something happen, it is applied to reshuffle information already contained in a system. Paraconsistent logic does not even have to be applied to inconsistent systems, and even when it is, derivable contradictions do not have to be interpreted as "true". What we need is not logic but semantics, although ...


7

There is a lot of writing both in favor and against AC from a philosophical standpoint - e.g. in favor see Penelope Maddy's Believing the axioms. However, there are also more mundane issues. I think that, whether or not it's ideal, a key point here is usability. An answer like this may seem dubiously appropriate at philosophy.stackexchange, but I think it'...


6

An Intuitive Walkthrough of PA as a formal system *Peano Arithmetic are a set of axioms in first order logic that describe how arithmetic of the natural numbers works. A first order formal language is a collection of variables, constants, logical symbols (such as negation, conjunction, etc.), parentheses, function letters, and predicate letters. Function ...


6

Quine did not specifically study "higher set theory", and his positions on the issue are mostly generalities following from his empirical holism (mathematics is the "entrenched" part of the "web of belief" that touches on experience at the observational boundaries) combined with the indispensability argument (what is indispensable in empirical science, e.g. ...


6

Is the axiom of infinity truly an axiom? Yes, it is an axiom of set theory. But in mathematics an axiom of a theory does not have to be plausible according to our everyday intuition. The only requirement it has to satisfy: The axiom does not contradict the other axioms of the theory. Of course axioms should not to be chosen arbitrarily. They should serve ...


5

I do not see any similarity between Kant's noumenon, taken as the thing-in-itself, and the empty set from mathematical set theory. Kant and science in accordance with him hypothesize the existence of objects in the real world. According to a constructivist epistomology we do not have direct access to these objects (thing-in-itself). But that hypothesis is ...


5

gnasher729 raised an important point that deserves some expansion: "In formal logic, implication x ⇒ y and equivalence x ⇔ y are very obviously useful - they directly express the possibly most important concepts of formal logic." The main point that I want to bring up is this: Naive set theory is not the only important set theory, and Classical logic is not ...


5

In Second-order Logic, the comprehension schema (considering for simplicity only unary predicate variables) is: ∃X∀x [ ϕ(x) ↔ X(x) ], where x is an individual variable, X is a 1-ary predicate variable and X may not occur free in ϕ. What prevents form generating Russell's Paradox ? Two facts: (i) we cannot substitute X for x. A s-o language for sets ...


4

Since the mathematical point has been made above, I'll just comment on the ontological side: One could ask the same question about everything, from the number 1 to human beings. Is there just the one number 1, or are there many isomorphic mathematical objects with its properties? Is there just one me, or are there many other isomorphic (upon some ...


4

The answers above give you the mathematical reason within ZFC for the uniqueness and existence of the empty set. For this from an intuitive point of view, you can use the analogy with a box. A set is not a box, but the content of the box. So, you can have two different empty boxes, but their content is the same : the "empty content".


4

There is only one empty set. Two sets are considered as different by ZFC if one contains an element not within the other. This comes from the extensionality axiom of ZF.


4

Is (verb : to be) is a simple word with many menaings... From a "mathematical" point of view, we can identify three different "contexts" : "Plato is a philosopher"; this context is relative to an object (or individual) belonging to a set or class. In modern math (set theory), this is expressed as : Plato ∈ philosphers. "a man is a male"; this context is ...


4

This is just part of Quine's naturalism, a sort of science-first approach to everything. That is mainly what underlies his suspicion of higher set theory. Here is Quine discussing the matter, from his book Pursuit of Truth (1990, pp. 94-95): Truth in mathematics What now of those parts of mathematics that share no empirical meaning, because ...


4

For category theory, see Natural number object. For set theory, see Set-theoretic definition of natural numbers. And see Category of sets for a link between the two theories. See also the post Categorical foundations without set theory and Philosophical Significance of Category Theory. From a mathematical point of view, the proposed construction of the ...


4

Throughout, I'm assuming that ZFC is consistent. There are a lot of confusing points here, and I don't really understand what you're setting up with bijections. However, I believe the key mistake you make isn't actually related to sizes of sets at all, but rather a serious misapplication of the incompleteness theorem. The language about infinite sets and ...


4

No, they do not partition states of affairs, and not only because compossibility isn't transitive. It is not a binary relation at all: any pair from x, y, z may be compossible, but not the three of them together. For example, take x "being a right triangle", y "being an isosceles triangle", and z " triangle having a 60° angle". Right or isosceles triangles ...


3

In general, the subformulas of Φ are those formulas used in the construction of Φ. You can see the construction of Φ as a tree, and every formula in that tree is a subformula of the resulting tree trunk (maybe this is more intuitive than the recursive formal definition you see in the book). An informal way to construct your formula and to show is well ...


3

I didn't know that this was... allowed, to extract information about the world from a formal system in this way. There's a tradition in Philosophy that uses mathematics to characterise physical ontology; starting with the Pythagoreans and continuing with Platos Timaeus; in the contemporary modern world, Physics - what was once Natural Philosophy - describes ...


3

There are two possible logical functions with no inputs, "true" and "false". We don't use symbols for them, just names. There are four possible logical functions with one input: "True" and "false" (whatever the input is, the output is "true", or whatever the input is, the output is "false"), identity (output = input) and negation (output = opposite of ...


3

I think your argument is essentially correct, but that you have to take care in how you describe the universe. Do you mean everything that is, or everything that ever was? Or, bearing relativity in mind, are you interested not in matter but in the collection of events which have happened throughout space-time? Consider a simplified account of the universe ...


3

A red-haired man is male, is a mammal, has two legs (usually), has red hair. Think what you can do with red hair: You can cut it, you can colour it, you can pull it out. Can you do any of these things with a red-haired man? You can "cut" him, but that means something different (likely a brutal knife-attack) than cutting hair. You can "colour" him, but that ...


Only top voted, non community-wiki answers of a minimum length are eligible