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Nausea by Jean-Paul Sartre.
The Prince by Machiavelli.
Categories by Aristotle.
A bit longish, though if this is a single take, I'm impressed by the presentation (I've yet to see a cut in the first 15 minutes or so).
"Similarity" and "difference" are relations, and as the speaker notes, relative to viewpoint. I've done some earlier research and thinking into the meaning and distinction of value, where utility-value (related to economic value) is based on relations as well -- to the producer or creator of an item or service, as well as the user or owner's use value. And these are not generally intrinsic.
Aristotle's Categories are a good starting point for considering relations, even now.
As you can see, there are far more fundamental axioms than the one that the Objectivists posit and even these can be reduced further. The choice of which first principles to choose are by no means an objective one, but instead a pragmatic one. Miss Rand chose one which happened to be convenient for supporting her preconceived notions, as did Marx, Aristotle, and Descartes.
That is distinct from faith (in the proper sense), which is belief in something without any evidence--that is, belief disconnected from reality. True faith is always immoral.
You're establishing a useless distinction: If you told my about a Faerie you saw one day and I believed you, then my belief would not be without evidence; your testimony counts as evidence. Doubtless, you consider that to be insufficient evidence, but it is evidence non-the-less. You are positing a 'faith' that no one, religious or not, holds.
Not to put to fine a point on it, but what you seem to mean is that faith, as you define it, is a belief based in something other than Objectivism, which is also a useless (and subtly religious) definition.
I'm going to have to stop this discussion here because this tells me that your entire argument rests on an univocation. I really don't have the time to discuss this topic to the length that it deserves so I'm just going to have to drop this. If you want further reading, you can try the Categories by Aristotle.
The cataloguing system uses a hierarchical classification, based on one originally developed by Thomas Jefferson, on whose initial donation the Library of Congress is based. This is known as the Library of Congress Classification, and is used to specifically locate a given title or work within the stacks, that is, each item has one and only one location.
There are also subject headings which are more tag-based, though also on a controlled vocabulary. A given work is given a (relatively small number) of subjects to which it's associated. These are not hierarchical, though of course the listing of subject headings itself follows a sequence. Unlike the classification, which assigns a single location to each work, the headings are a search aid to patrons searching for a set of related works within a subject heading, or facilitate branching of a search to possibly related subjects.
Tagging systems, especially ad hoc tags supplied by untrained users, are popular but tend to produce numerous issues over time. Not that formal systems (as with the LoC systems mentioned here) are immune to same. One feature of the LoC systems is that they've evolved processes for managing change over time. Examples would be terminology or classifications which are now deprecated, or of regions and polities which have changed or no longer exist (e.g., the Austro-Hungarian empire, the USSR), or of changes in underlying classifications (e.g., of chemical elements or of biological classifications, both of which have evolved significantly over the life of the Library of Congress).
The history of hierarchical information classifications is long and IMO fascinating, dating at least to Aristotle and his Categories, as well as numerous variants used in classifications of knowledge (such as Francis Bacon's) or encyclopedias, including Diderot's and Britannica.