Is it provable that all isomorphic structures share the same properties? This is my first semester as a math student and among my classes I'm taking calculus. The book the course is based on is Calculus by Michael Spivak. This book is amazing and since it starts with the properties of real numbers I'm also reading the end of the book where he says that such a structure is unique up to isomorphism. At that point this is what the book says: 

... any significant mathematical property of the real numbers will be true
  for all isomorphic fields. To be candid I should admit that this last assertion is just
  a prejudice of the author, but it is one shared by almost all other mathematicians.

My teacher told me that this assertion actually has to do with logic and that I shouldn't worry too much about it. The problem is I don't want to stop there and want to learn more. So, my question is if that Spivak's assertion is provable or why he is saying that it's just a prejudice. If it's provable I'd like to know what tools I need to have in order to understand it and also it would be great if someone can give me the general idea behind it and some references. 
 A: Welcome to mathematics! 
Isomorphism is a relative concept: as you make progress in your studies you will see that things are said to be isomorphic as "this" or "that". Every time the word "isomorphism" comes up, it will be isomorphism of a specific kind of objects. The word itself means that the structure of the two objects is one and the same. For example, two vector spaces are said to be isomorphic (as vector spaces) when there is an bijective linear map between them. In that case, every property which can be described by the vector space structure can be carried from each of them to the other. The map itself is called a "vector space isomorphism".
Concerning the real numbers, their whole structure stems from their structure as an ordered field: their "shape", their "algebraic structure". Thus, what Spivak means, or rather, my interpretation of what he writes, is that if something is isomorphic to the real numbers as an ordered field, then the two share the same ordered-field-properties, which in turn means that you can "copy and paste" the whole structure of the real numbers on that very object.
I hope my answer shed some light!
