TITLE: With this definition of completeness, Gödel's Incompleteness result seems not surprising, so why it was back then?
QUESTION [21 upvotes]: According to wikipedia a theory (i.e. a set of sentences) is complete iff for every formula either it, or its negation, is provable.
On the other side, a logic is complete iff "semantically valid" and "provable" are the same.
The first notion of completeness is with what Gödel's Incompleteness result is concerned, but then I do not understand the significance given to it, or why it surprised people? Because if I read the first definition, if I can give a formula which is satisfied in one model, but not the other, then this formula and its negation could not be provable (if the logic is sound). And in general, I would expect this property more than the property that in every theory, every sentence either holds for all models (is valid), or its negation is valid.
To be more specific, Gödel's result in its original formulation is concerned with Peano arithmetic, but it also holds in some form of first order theory of the natural numbers with multiplication and addition as primitive notions, and for this we know that the natural numbers are not the only model.
So, why did it come as a surprise? Did people really think that for every theory and a given formula, either it or its negation are semantically valid, i.e. fulfilled by every model?

REPLY [10 votes]: Godel's incompleteness proof is published in 1931.  In 1931, the mathematical community's response to your question 

"Did people really thought that for every theory and a given formula, either 
  it or its negation are semantically valid, i.e. fulfilled by every model?" 

and comment 

"Because if I read the first definition, If I can give a formula which is satisfied in one model, but not the other, then this formula, and its negation could not be provable (if the logic is sound)."

is "What's a model?"
Some mathematical logic from before the 1950s was subsequently recognized as model theoretic.  But model theory wasn't a thing until Tarski and his students fleshed it out in the 1950s and 1960s.
The setting of Hilbert's extreme optimism is described in some detail in Hilbert's Philosophy of Mathematics, especially section 5, "Hilbert's Programme and Logical Positivism".

Added in edit, 20180407.
Skolem for instance did not interpret either of his Lowenheim-Skolem Theorems (the first, with AC and the second, without) in a model theoretic sense.  I was first exposed to this in [G], 

"From our perspective, it is obvious that the possession of set-theoretic properties, if relative to anything, is relative to the model one is using to interpret the formal system.  Several writers have noted that this was perhaps not obvious to Skolem who, they claim, often interpreted the LST as demonstrating the relativity of set-theoretic notions to the axiom system itself  (see, e.g., his 1929b, 293 [sic]; and 1958, 635-637)."
  Those references are 


1929b 'Uber einige Grundlagen fragen der Mathematik', Skrifrer, Vitenskapsakadernie I, 4, 1-49; reprinted in Skolem 1970, 227-273. 
1958 'Une relativisation des notions mathematiques fondamentales', in Colloque: nationaux du Centre National de Recherche Scientifique, Paris, 13-18; reprinted in Skolem 1970, 633-638.
1970 Selected works in logic, edited by Fenstad, J.E., Oslo (Universitetsforlaget).

In short, when we say the Lowenheim-Skolem theorem is a theorem about models, we are very much putting words into Skolem's mouth.  He writes as if the axioms pick out a unique model, so the study of models would be redundant.  He also writes as if the existence of non-standard models of arithmetic are a defect of the underlying logical system not of the proposed model axiom systems.  See 

Skolem, "Uber die Unmoglichkeit einer vollstandigen Charakterisierung der  Zahlenreihe mittels eines endlichen Axiomensystems. Norsk Matematisk
Forening, Skrifter, pages 73–82, 1933. (reprinted in Skolem 1970, 345–354.)
Skolem, "Uber de Nichtcharakterisierbarkeit der Zahlenreihe mittels
endlich oder abzahlbar unendlich vieler Aussagen mit ausschliesslich Zahlen-variablen.  Fundamenta Mathematicae, 23:150–161, 1934.  (reprinted in Skolem 1970, 355–366.)

It's easy to look back in time and frame the work of others in terms that were not available and would have been alien to them.  When we push the ideas and language of model theory earlier than the mid-1930s, we are guilty of doing so.  And even then, once such terms become available they are not instantly available to all practitioners; it can take more than a decade for the good ideas to be communicated widely and explained clearly.
[G]  George, A., "Skolem and the Lowenheim-Skolem Theorem: A Case Study of the Philosophical Significance of Mathematical Results", History and Philosophy of Logic, 6 (1985), pp. 75-89, http://www.fitelson.org/140A/george.pdf .