VazScep

Uh...the Stone Cutters?

Okay. Now what about the what you've written in between the apostrophes. The phrases "ALL n members" and "Collection of infinitely many finite elements" are not sentences, let alone propositions. Any even if they were (‘ALL n members’ AND ‘Collection of infinitely many finite elements’) AND (‘Successor’) --> True would be a tautology. Any proposition materially implies any true proposition. This fact is often stated as an axiom of propositional calculus: P -> (Q -> P) Rather than trying your hand at logical symbolism, why don't you try straight forward English.

No idea what this means. Is "-->" implication?

By the way Doron, do you realise that every single use of a quantifier in ZF set theory ranges over a domain which has no cardinality, namely, the class of all sets?

But to say that, whatever natural number you choose, that number will be non-negative, is to say something about all natural numbers. But it does not say anything explicit about the set N. I'm not sure I understand what you are saying, but here goes: "The set N" does not mean the same thing as "all N". Statements involving "all N" are statements about the individual natural numbers, to be translated as I suggested as "whatever n in N you may choose". However, statements involving "the set N" do not have to say anything about natural numbers. For instance, I can tell you that N belongs to the set {N}, without having to discuss any given natural number. Cardinality is a property of a set defined in terms of mappings, which themselves are defined in terms of individual members of the set.

Actually, forget my attempt at a grammar lecture. I have a feeling both versions would be criticised by an English professor. Since these idioms are exclusive to mathematics and logic (we do not say "for all people, it is free", nor "for each person, it is free), I would say that both phrases are intended as a shorthand for "whatever person you choose" or "given a person" etc...

"For each..." does not convey the same meaning as "There exists some abitrary...". Existence statements only guarantee a single object satisfying a condition. I have no idea what it means for an arbitrary object to exist. Is the number 3 arbitrary, for instance? Arbitrary in what sense? By the way, every time you read "For all..." you should be able to replace it with "For each...". If you only have a complaint with the former, why don't you try doing this?

Synonyms are interchangable in a sentence. "For all..." cannot be replaced by "For complete..." Are you suggesting that a sentence such as "All humans are mortal..." is actually to be understood as "The 6 billion (or so) humans are mortal"? And likewise with other such statements? Well the class of men is not mortal, so yes, the property is not changed by the quantity of men. Which definition? I don't understand. "There exists a man who is mortal" is not what we want, so what does our statement have to do with "there exists..."? I'll stop here. The rest confuses me just as much. By the way, if the logical idioms of Hebrew drastically differ from those of English then we may have serious communication problems here.

While this was observed, the interesting thing is that Cardano's method of solving cubic equations with real solutions sometimes required an intermediary use of complex numbers.

The way we view and do mathematics has changed over the centuries, and so mathematical names which are retained for the sake of tradition are often misleading. IIRC, the term "imaginary" was coined by Descartes, when complex numbers had not long appeared on the scene as a "cheat" for solving cubic equations, and when their legitimacy was very much in doubt.

Can you state a theorem and provide its proof, appealing to nothing but your axioms, to such a sufficient level of rigour that I can substitute the principle terms with random words, follow your inferences, substitute back and still end up with your conclusions?

Can you state a theorem and provide its proof, appealing to nothing but your axioms, to such a sufficient level of rigour that I can substitute the principle terms with random words, follow your inferences, substitute back and still end up with your conclusions?

Please prove one theorem from these two axioms, or provide additional axioms which allow you to prove a theorem. I cannot deduce anything of interest from them as they stand.

Please prove one theorem from these two axioms, or provide additional axioms which allow you to prove a theorem. I cannot deduce anything of interest from them as they stand.

Just to add: The reason zero took so long to become an accepted part of mathematics is because most of the mathematical traditions in history have viewed geometry as the subject's foundation. While number theory and arithmetic were certainly studied, definitions and proofs in these fields were usually put in geometrical terms. For instance, the Babylonians, the Muslim mathematicians and the later western European mathematicians would express a simple quadratic equation such as [math]x^2+2x=3[/math] as "a square plus twice its side is 3," and a solution for x would be found by a geometrical construction*. Consequently, zero and negatives were not developed (the work of Brahmagupta was ignored by later mathematicians), not having obvious geometrical interpretations. Their inclusion as legitimate mathematical concepts from the 17th century represents a conceptual shift away from a view that "all is geometrical" to a more arithetically-oriented view. It is also interesting to note that complex numbers were introduced at around the same time, though the secure position they enjoy nowadays as legitimate concepts had to wait for Hamilton and others in the mid-19th century. *In this case, you construct a square with side x, extend the edges by 1 unit, and then, literally, "complete the square" by adding a smaller square of side 1 in the corner, to deduce that [math]\left(x+1\right)^2=4[/math], and thus that [math]x=1[/math]. EDIT: Just trying out LaTeX in this forum.

Just to add: The reason zero took so long to become an accepted part of mathematics is because most of the mathematical traditions in history have viewed geometry as the subject's foundation. While number theory and arithmetic were certainly studied, definitions and proofs in these fields were usually put in geometrical terms. For instance, the Babylonians, the Muslim mathematicians and the later western European mathematicians would express a simple quadratic equation such as [math]x^2+2x=3[/math] as "a square plus twice its side is 3," and a solution for x would be found by a geometrical construction*. Consequently, zero and negatives were not developed (the work of Brahmagupta was ignored by later mathematicians), not having obvious geometrical interpretations. Their inclusion as legitimate mathematical concepts from the 17th century represents a conceptual shift away from a view that "all is geometrical" to a more arithetically-oriented view. It is also interesting to note that complex numbers were introduced at around the same time, though the secure position they enjoy nowadays as legitimate concepts had to wait for Hamilton and others in the mid-19th century. *In this case, you construct a square with side x, extend the edges by 1 unit, and then, literally, "complete the square" by adding a smaller square of side 1 in the corner, to deduce that [math]\left(x+1\right)^2=4[/math], and thus that [math]x=1[/math]. EDIT: Just trying out LaTeX in this forum.