Knowledge itself is more than meets the eye, make no assumption, recognise the faults in the faculties and learn who and what not to follow.
By Jack Don McLovin - Multidiscplinarian -- Jack Has Been writing pages a day for the past 10 years under varying anonymous accounts, always drifting from place to place trying to gain popularity in a group, and start again, and repeat.
First Order Logic, or Predicate Logic is the foundation of (almost) All Mathematics, and therefor the ideas of Objectivisim, Concreteness, Systems, Hierarchies, and Science.
First order logic relies on certain axioms being true, which are generally accepted as a given before beginning the reasoning towards a conclusion which is what allows the axioms to extend to granting the idea of proof.
It includes the following Axioms:
The axiom of extensionality: This axiom states that two sets are equal if and only if they have the same elements. In other words, if A and B are sets, then the proposition "A = B" if and only if "for all x, x is an element of A if and only if x is an element of B" is always true.
The axiom of identity: This axiom states that every object is identical to itself regardless of how we refer to it. In other words, for any object x, the proposition "x = x" is always true.
The axiom of equality: This axiom states that if two objects are identical, then they have the same properties. In other words, if x and y are identical, then for any property P, the proposition "P(x) if and only if P(y)" is always true.
The axiom of comprehension: This axiom states that for any property, there exists a set of all objects that have that property. In other words, for any property P, there exists a set S such that "for all x, x is an element of S if and only if P(x)" is always true.
The axiom of uniqueness: states that for any set A and any property P, there exists a unique set B that contains exactly those elements of A that have the property P.
The axiom of existence: there exists at least one object or set.
The axiom of extensionality: this is not true because it as it is defined and used applies to all sets equally, whereas really it only applies to unordered sets, and it also doesn't apply in the numerological composition of words, which exists in every language.
The axiom of identity: This is incorrect, because it assumes that framing and phrasing don't have a measureable effect on the world, such as when people prefer to buy things that are affordable rather than cheap, or they prefer a voucher over a discount even if the value is equal.
The axiom of equality: It says that for any identical objects their properties are equal, but this is not true because the same colour blue can appear different to different people.
The axiom of comprehension: This is incompatible with the axiom of uniqueness, because it says that if an object satisfies a given condition there is only one of it, whereas the comprehension axiom says "the set of all objects that satisfy a given condition", which implies there is more than one.
The axiom of uniqueness: This is incompatible with ZFC (the extension of FOL that applies to most math) because an axiom of ZFC is the Axiom of Repetition which is that you can have a repeat of any set, whereas the uniqueness axiom says there is one and only one of any given set.
The axiom of existence: The way this is stated is incomplete because the way it is used is that there exists at least on SEPARABLE object or set, but this is not true because no object or set is, has been or ever will be separable from it's environment. The definition of an Object is something that can "be referred to or quantified", and it is subcategorised as either Concrete or Abstract, but both of these are categories of each other / perceptions of the same thing. Because the idea of something being concrete is itself abstract, and defining something abstractly can only be done with concrete forms.
It's not an issue to say that first order logic has it's problems, but that doesn't mean we are to dissuade from using it, only to prepare a new way that goes alongside it. As the axioms themselves are inherently assumed unquestionably. And yet here we have some valid critiques. And so the issue arises that we need new axioms that can be moved through outside of First Order Logic