Hm, maybe the author wants to confuse his students as much as he can, for whatever reason (maybe he has a trauma and hates students or is simply evil?).

The standard reading is that x+4=7 is an equation with one unknown to be solved. There are three possibilities: (i) it has at least one solution, (ii) it has no solution, (iii) it is undecidable within the used set of axioms. In case (i) you may have two subcases (ia) there's a unique solution, (iib) there is more then one distinct solution.

In this case, interpreting as ##x## standing for any real number we know that this is linear equation with one unknown, which has a unique solution, and indeed subtracting 4 on both sides of the equation tells you that without other possibility you must have x=3. So there's exactly one solution of the equation. Of course, putting for x any other number than 3 leads to a wrong equation.

Yes, exactly what I thought. The "open sentence" thing seems to be an unnecessary additional level of abstraction -- I don't see what it buys beyond the "standard reading." That's why I wondered about the vintage, 1966, right in the "new math" era, which to me seemed to be about making simple ideas as abstractly confusing as possible.

As mentioned above, this has nothing to do with calculus and little to do with the OP in this thread; I will let it be and refrain from further distraction.

My suggestion, if you have access to a library , is to browse through the Calc section and try to get a feel for which book is best for you. Spend a few minutes browsing through before making a decision and then buy the one that feels best for you.

I won't say it is a redundant term because it is a term from Formal Logic. My guess is, author introduced it so that readers can gain some mathematical maturity. Most math books have a chapter on Logic/Set-theory so readers can write mathematical proofs. Analogous to how physics books have a chapter on mathematical methods.

You have to read it in context. The author explains what it means for something to be "well defined." He further goes on to show that P implies Q, is not the same as Q implies P, etc. Furthermore, he takes this argument further and explains how, if P is false we do not care wether Q is true or false. You know, conditional statements etc....
The author's intent is to show that mathematics is a very precise language, and ambiguity has no place in it!

Keep reading the book. The intro section is weird if you are not used to this thinking. A few pages more and you see what a Field is...

@mathwonk, are you familiar with this book? If you are not, I think you would find it quite interesting and lot to be gained from it. I think its worth having on a book shelf next to Courant.

I suggest you look at how he builds integration and diff. It is really beautiful how Moise built it up. Some geometry, explanation of coordinates, what a right hand system is, what the secant to a line is, how we need to redefine a tangent from our experience from Euclidean Geometry, proof by induction, nice and intuitive explanation of the well ordering principle. How the tangent of a circle can be considered a special case of a more general...

Concept of Area. How "Area" is really ambiguous

He even explains how Dedekind cuts relate to two other concepts...

well i do not have access to the book by moise but i did get to read the preface of the 2nd (and 1st) edition. This is the most clear and persuasive statement of philosophy of teaching and writing a calculus book, plus how to learn that I have seen in any textbook. of course moise is a world class mathematician which always helps. this is not a parallel of spivak or apostol. those are thorough going theoretical books. this book has been written to be useful as both a theoretical and a less theoretical book, by putting the harder theoretical parts at the end of the sections or chapters, so they can be skipped withut losing grasp of the facts to be used. he also tries to convey, even when omitting the theory, both computational techniques and intuitive concepts. If he succeeds in his stated goals, this should be an excellent book. It thus seems to have more in common with courant in its goals and approach. there is however a reason that the other 3 books, apostol, spivak and courant, have been almost univerally recommended for so long. in particular i have never heard anyone call apostol "overrated". anyone who says this would have to go a long way to avoid being considered a very unreliable source in my opinion. although an intelligent person might be able to make a case somehow, i do not know what it would be based on. i would say apostol is a strong candidate for the absolute best calculus book of all, but only for the very strong and serious student. spivak is also superb, and is more "fun" than the somewhat sombre apostol, but spivak is only for the future pure mathematician. apostol also offers more applied ideas, and also does several variables and linear algebra. still i second the opinion that the best way to proceed is to sit in the stacks in the calculus section of a university library and do some reading.

the only book by moise i have read more of is the one on geometry. while very scholarly, rigorous, and correct, i found it unappealing to read for a student. apparently the calculus book was written with more care in readability. but as i said i have not been able to read it, at least not lately. i seem to have looked at it in the past in my career as a teacher, and it apparently did not leave me wanting to own one. maybe that was in the period when i could not afford books, or maybe i just had too many already, or maybe it did not have any content that i myself felt i needed another source for. spivak always has something that i can learn from. e.g. there is a problem that shows how to prove a function with derivative zero is locally constant, without using the MVT, which not everyone knows how to do. someone asked me this question just a few days ago on the professional math site "mathoverflow". perhaps unfortunately, i gave away my copies of both spivak and apostol a few years ago, to the undergraduate math dept library.

If anyone feels benefited by spivak's book, i would like to remind them that he is still alive and earns his living primarily from sales of it.

May be, it's due to the "new math movement" (or however you call this nonsense). Sometimes, I have the impression that the purpose of mathematics and physics didactics is to deform the subject such that even an expert cannot make any sense about it anymore, let alone the poor students who are exposed to these ideas.

At highschool we had to use a textbook, where they considered it pedagogically better not to write down the differential in integrals, i.e., instead of writing
##\int \mathrm{d} x x## they wrote ##\int x##. I immediately told the teacher, that I'd not use this nonsensical notation since it's not leading to the correct dimensions to begin with, and that you cannot know wrt. which variable you should integrate. After a short thought the teacher said that I am right and regretted to have to use this strange textbook ;-))).