Originally Posted by Mustafa16
The question is quite tricky, because it depends on what exactly the terms true
, and proof
mean. One of the greatest breakthroughs in mathematics in the 20th century are Kurt Gödel's incompleteness theorems:
Theorem 1: In every sufficiently complex axiomatic system there are statements that are true but not provable.
Theorem 2: If a sufficiently complex axiomatic system claims its own consistency, it is inconsistent.
You will find these results in Kurt Gödel's original 1931 paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems I.
as Theorem VI
and Theorem XI
. Generally spoken, the values true
, and falsifiable
(=testable) are distinct truth statuses in an algebraic lattice that successfully implements the absorption law. They are not the same. This surprisingly corresponds with the classification in Immanuel Kant's Kritik der reinen Vernunft
- true: analytical a priori
- provable [math]: synthetic a priori
- testable (=falsifiable) [science]: synthetic a posteriori
Therefore, the question What proof is there?
is literally a request for an axiomatic system in which you can derive the existence of a first cause from basic statements that are actually unrelated. Note that provable
never means that the basic statements would be true. Axioms have no truth status at all.
But then again, such axiomatic system does indeed exist. In his seminal work Physics
, Aristotle derives that there is exactly one attraction point at the origin of the universe for the repeated application of the causality function.
Aristotle, Physics, book VIII, part 5: If then everything that is caused must be caused by something, and the cause must either itself be caused by something else or not, and in the former case there must be some first cause that is not itself caused by anything else, while in the case of the immediate cause being of this kind there is no need of an intermediate cause that is also caused (for it is impossible that there should be an infinite series of causes, each of which is itself caused by something else, since in an infinite series there is no first term)-if then everything that is caused is caused by something, and the first cause is caused but not by anything else, it much be caused by itself.
Aristotle, Physics, book VIII, part 6: It is sufficient to assume only one cause, the first of uncaused things, which being eternal will be the principle of causality to everything else.
The belief in a first cause is therefore equivalent to the belief in the  finitude of time,  generalized causality,  strictly-ordered consequentiality, and  the impossibility of supertasks. Any axiomatic system that rests on these four axioms will automatically derive that there is exactly one first cause for the universe, located at the origin of timespace. Note, however, that this statement is only provable
in such underlying axiomatic system.
means something entirely different. In terms of Tarski's undefinability theorem
, pronouncing a true
statement in a particular sufficiently powerful object language amounts to expressing one of its own construction theorems that would otherwise be expressed in its metalanguage. Such statement is always true but not provable
in the object language. In fact, a statement cannot be simultaneously provable
in the same axiomatic system. That would cause an ambiguity in its underlying algebraic lattice and result in a violation of the absorption law. Hence, it is the one or the other.