Difficulty levels are assigned to "facts" -- not to terms.
|Difficulty level (numeric value, click on link to access list of facts with that difficulty level)||Qualitative description||Who should be able to get this?||Generalizability|
|0 (zero)||follows directly from the definition, to the point where the result may be called "trivial" or "direct" and may not merit a proof.||anybody who understands the definition.||statements are usually too trivial or obvious to even consider generalizing.|
|1||follows from the definition, but requires one to work out steps, basically using a "follow your nose" approach. Note that in case of multiple alternate definitions, it may require picking the right one, and the proof may be considerably harder if one chooses the wrong definition. Any facts used are facts that automatically appear while following your nose.||anybody who knows the "correct" definition to use and has enough practice with mathematical reasoning should be able to get it. No deep understanding of the objects involved is required, because the proof is very superficial.||such statements are also likely to be generalizable (possibly in multiple ways) as they are less "contingent" on specific structural aspects.|
|2|| requires one somewhat novel idea and does not just involve "following your nose." For instance, it may require picking a particular aspect of the definition, or using one specific theorem or characterization, or constructing something specific.
In addition, easy counterexample searches usually land at this difficulty level, in so far as they require some degree of experimentation.
|This typically requires somebody who has an intuition into the relevant ideas from group theory. Superficial formal reasoning alone is likely to fall short of discovering the relevant ideas.||such statements may generalize but the generalization needs to be done carefully in a tailored fashion so as to preserve the proof specifics.|
|3||requires either one fairly difficult idea or a synthesis of multiple ideas. This difficulty level includes many of the foundational results that substantially improve our understanding of a specific structure.||discovering the proofs of results at this level could be quite difficult and even seasoned researchers may struggle for days. However, understanding the proof should be easy for most people who are able to discover difficulty level 2 proofs. Might have been a major theorem of a paper around 1900 or an important lemma of a paper around 2000.||Generalization is highly unlikely to prior generalized structures. Often, such statements may motivate new definitions with the explicit goal of generalizing the fact.|
|4 and higher (5, 6, 7)||requires a fairly complicated synthesis of multiple ideas. Results at this level may be foundational, but may also be "too hard to be foundational." Often associated with very complicatedly formulated lemmas that appear in the proofs of very hard theorems.||Similar to difficulty level 3, but harder.||very unlike to generalize.|
Pages using the property "Difficulty level"
Showing 25 pages using this property.