Disa nga këto parime mund të dëshmohen se janë pasojë e parimeve të tjera. Shumëllojshmëria e formulimeve të këtyre parimeve aksiomatike lejon një analizë të hollësishme të formulimeve të kërkuara në mënyrë që të nxjerrin rezultate të ndryshme matematikore.[14]

Sistemet e Fondacioneve të Reja të NFU (duke lejuar urelements ) dhe NF (që nuk kanë ato) nuk bazohen në një hierarki kumulative. NF dhe NFU përfshijnë një "tërësi të gjithçkaje", në lidhje me të cilën çdo grup ka një plotësues.

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Një issue i njohur është pyetja normale e hapësirës Moore, një pyetje në përgjithësi topologjia që ishte objekt i një studimi intensiv. Përgjigja për pyetjen normale të hapësirës Moore përfundimisht u vërtetua se ishte e pavarur nga ZFC.

Këto veti zakonisht nënkuptojnë se numri kardinal duhet të jetë shumë i madh, me ekzistencën e një kardinali me pasurinë e specifikuar të paprovueshme në teorinë e vendosur Zermelo-Fraenkel.

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Shumë veti të grupeve Borel mund të vendosen në ZFC, por provimi i këtyre pronave të vendosura për grupe më të komplikuara kërkon aksioma shtesë që lidhen me përcaktimin dhe kardinalët e mëdhenj.

Teoria e vendosur fillon me një lidhje themelore binare midis një objekti o dhe një grupi A Nëse o është një anëtar (ose element ) i A, përdoret shënimi o ∈ A.

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