Is elementary algebra consistent?

Question

for background I am a high schooler. I know some higher math (complex analysis, group theory etc) but it seems I am fairly out of my depth in this question.

I have had a nagging question that entered my mind a while back that I hope someone on here can help me with. I have being thinking about elementary algebra and its "consistency" -- after googling I think that is the right word.

It seems that everything hinges on our elementary algebra rules not conflicting, but I can not think of a way to make sure they do not. Is there some proof to ease my mind?

I am also equally curious about this same question but for geometry.

Also I was wondering

if I can define geometry by algebra (and coordinates) would showing algebra has no contradictions be enough to show geometry has none?

If there is some kind of proof I would be very interested in knowing more, so resources on that kind of thing (showing arbitrary sets of axioms do not make contradictions) would be much appreciated.

Hopefully I was able to properly ask my question, It is hard to find the right words when I know so little about it.

Answer 1

I appreciate your curiosity. In regard to the information you give (that you are a secondary school student, etc.), I'd recommend you (I suspect, as second to none) Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid in order to form a firm conceptual basis (which is indispensable not to get lost on the way). Aiming at a popular audience, the discussion roams over a broad range of topics the author intertwines. The following excerpt (p. 94) may suggest that it covers also the subjects you look for:

A full formalization of geometry would take the drastic step of making every term undefined—that is, turning every term into a "meaningless" symbol of a formal system. I put quotes around "meaningless" because, as you know, the symbols automatically pick up passive meanings in accordance with the theorems they occur in. It is another question, though, whether people discover those meanings, for to do so requires finding a set of concepts which can be linked by an isomorphism to the symbols in the formal system. If one begins with the aim of formalizing geometry, presumably one has an intended interpretation for each symbol, so that the passive meanings are built into the system. That is what I did for p and q when I first created the pq-system.

But there may be other passive meanings which are potentially perceptible, which no one has yet noticed. For instance, there were the surprise interpretations of p as "equals" and q as "taken from", in the original pq-system. Although this is rather a trivial example, it contains the essence of the idea that symbols may have many meaningful interpretations—it is up to the observer to look for them.

We can summarize our observations so far in terms of the word "consistency". We began our discussion by manufacturing what appeared to be an inconsistent formal system—one which was internally inconsistent, as well as inconsistent with the external world. But a moment later we took it all back, when we realized our error: that we had chosen unfortunate interpretations for the symbols. By changing the interpretations, we regained consistency! It now becomes clear that consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it. By the same token, inconsistency is not an intrinsic property of any formal system.

As for the style of the book, these words from a review give a fair description:

Professor Hofstadter's presentation of these ideas is not rigorous, in the mathematical sense, but all the essential steps are there; the reader is not asked to accept results on authority or on faith. Nor is the narrative rigorous in the uphill-hiking sense, for the author is always ready to take the reader's hand and lead him through the thickets.

The book has been translated, published and read all over the world since 1979. You can freely read it at or download its pdf from archive.org. I suppose one can get a second-hand in-good-condition copy of it at quite an affordable price today.