I have a question about axioms and implication: I just stumble upon various texts that use these concepts. like:
Mathematicians have always looked for axioms in order to find the fewest and simplest statements that imply all the other statements.
or
Hilbert included an axiom in his The Foundations of Geometry that implies Dedekind's axiom.
or
We show that in ZFA, set theory without the Axiom of Choice weakened to allow "atoms", Ingleton's axiom does not imply the Axiom of Choice.
or
Huntington showed that the three axioms together imply the axioms of Boolean algebra.
and others like these.
I wanted to know when implies is used in these kinds of sentences, Is the word "implies" means like implication in logic or it has a different meaning?
I could have asked this question on other forums but I thought you would know better.