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I'm trying to get an overarching understanding of the components of mathematical systems so that in my self study of each category of math I can break them down by their unique aspects, i.e. the operators they use, the major concepts they deal with (i.e. how calculus is about "change"), etc.

As far as my experience with formal math terminology goes, im rather weak, and i get utterly confused by the technicality required in formal definitions.

As a good starting point, I'd like to better understand what the difference is between an axiom, a theorem and a postulate. At my current level of knowledge i would use them interchangeably (lol), however I'm sure one is founded upon the others.

If someone could explain the logical hierarchy/relation between these three it would be greatly appreciated.

user1299028
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  • Theorems are proven on the basis of already proven statements (other theorems) or axioms while axioms (or postulates) are statements that are accepted as true without a proof. – Alessandro Codenotti Mar 26 '14 at 06:26

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If mathematics were a chess game, propositions are the possibile chess positions. Inference rules are the valid moves. Postulates (or axioms) is the initial position of pieces. Theorems are the positions you can reach in a game by applying moves to the initial position.

Emanuele Paolini
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  • So then axioms are the most fundamental "self-evident" principles, and through a series of inferences deemed valid we can deduce theorems from first principles? I'm still not entirely sure where propositions fit in then. When you say possible chess positions do you mean the entire range of possible board configurations or do you mean the possible attainable positions from one move to the next? – user1299028 Mar 26 '14 at 06:45
  • Proposition are all chess position in the metaphore, comprising position that cannot be reached in a real game. In mathematics they are all possibile statements that you can formulate, including false ones. – Emanuele Paolini Mar 26 '14 at 07:30
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Basically Theorems are based off of other proven statements such as Axioms or even other Theorems, and Axioms are kind of unchallenged rules and assumptions, such as simple addition equations.

zebragal20
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  • This question was already answered and has an accepted answer. You have not contributed anything new. Please refrain from doing so for very old questions. – Shailesh Jan 24 '16 at 15:27
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Axioms are the things that are taken as basic, unchallenged assumptions. Depending on what area of mathematics you are working within, these may change. For instance, all of the elementary results of arithmetic are usually taken as unstated axioms in higher branches of mathematics (so you don't prove $1+1=2$ in a calculus course, but might in a certain other courses).

Theorems are conclusions that can be drawn from a set of axioms by using the rules of logic. Not every such conclusion is bestowed with the title of theorem though: usually the conclusion has to be meritorious, and often the verification is non-trivial.

I've always thought of postulates as being slightly less fundamental than axioms, but nevertheless similar in their nature as assumptions from which other statements are to be proven. To give a natural language analogy: I take as an axiom is that I exist, and I postulate that my senses aren't being manipulated as part of a perfidious plot.

Rookatu
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  • So then is the difference between postulates and axioms a semi-arbitrary distinction about the relative technical degree of certainty which we can possess about them? Like, surely both axioms and postulates are "fact" insofar as they provide enough assurance of their veracity as one would usually be inclined to desire. But perhaps on a technical ontological basis axioms generally have a higher degree of provability than postulates? – user1299028 Mar 26 '14 at 06:52
  • One doesn't usually think about the "provability" of axioms, at least not within the system one is working. They are the substrate upon which everything else is built. Maybe a good analogy is that they are like the hardware of a computer, postulates are like the operating system, and theorems and lemmata are all the things one can do with that system. – Rookatu Mar 26 '14 at 07:46
  • To Euclid "axioms" were essentially basic statements about algebra or reasoning while "postulates" were statements about geometry. Modern mathematics makes no distinction between axioms and postulates. – user247327 Jan 24 '16 at 15:13