the facts of biology and mathematics are that every person has two parents, which means each of his ancestors also has two parents. Thus he should have:
8 great grandparents.
16 great great grandparents.
32 great great great grandparents.
64 great great great great grandparents.
128 great great great great great grandparents.
and so on.
Ten generations back there would be 1,024 great great great great great great great great grandparents.
Twenty generations back there would be 1,048,576 ancestors in the 20th generation.
Thirty generations back there would be over 1,000,000,000 ancestors in the 30th generation.
Of course the actually number of ancestors would be fewer because of many, many marriages between people who were cousins of some type, 4th cousins, 15th cousins, 21st cousins 5 times removed, 35th cousins, etc.
If you assume that the average length of a long series of Wizard generations would be between 20 and 40 years, then:
The tenth generation of a wizard's ancestors would be born about 200 to 400 years before him.
The twentieth generation of a wizard's ancestors would be born about 400 to 800 years before him.
The thirtieth generation of a wizard's ancestors would be born about 600 to 1,200 years before him.
So when did wizard society begin? How many centuries or millennia ago was that?
Who can keep track of all those ancestors?
In medieval and early modern times, European nobility usually tried to keep themselves a closed caste, separate from lowly commoners. Many noble institutions, such as the knightly orders, had rules requiring a certain minimum of noble blood.
Thus nobles compiled ancestor charts showing their 2 parents, 4 grandparents, 8 great grand parents, and so on back as many generations as a particular institution required for its members, and beside or above each ancestral name would be a picture of their coat of arms. The coat of arms would show the noble status of the ancestor - although many lowly commoners did and do have coats of arms.
Experts in European royal genealogy certainly have a hard time remembering all the ancestors in all the generations that the historical or legendary ancestry of various royal families go back to.
And they also know how many times a king or nobleman's wife had no recorded ancestry, so they can not tell whether she was royal, noble, or a lowly commoner.
For example, there is Agatha (before 1030-after 1070), the wife of Anglo-Saxon prince Edward the Exile (1016-1057), and mother of his children including Edgar the Aetheling (c.1051-c.1126), rightful heir of Anglo-Saxon England, and his sister St. Margaret (c.1035-1093), wife of King Malcolm III of Scotland (1031-1093) and ancestress of countless thousands of modern European nobles and monarchs. There are about a dozen different theories about which nobles, kings, or emperors Agatha might have been descended from, and I guess there isn't any actual proof that she was of noble and not common ancestry.
And they do know of a few commoners who became ancestors of European royalty and nobility, so they know that even though European royalty is sort of a separate caste, it is not 100 percent pure and separate from lesser mortals.
For example, Fulbert of Falaise was a tanner or possibly undertaker, whose daughter Herleva (c. 1003-c.1050) had an affair with Duke Robert I of Normandy (1000-1035) and became the mother of his son William (c. 1028-1087) before marrying Herluin de Conteville. William became Duke of Normandy and invaded and conquered England.
Bretislav I (1002/1005-1055) was Duke of Bohemia from 1035. He was the son of Duke Oldrich (c. 975-1034) and the beautiful but low born Bozena (died after 1052), daughter of Kresina.
And countless thousands of modern European nobles and royalty today can trace their ancestry to the commoners Herleva and Bozena.
What I imagine is that sometime in the past many of the higher and more prestigious wizard families decided that being pure blood was a desirable. And they got together and decided on some requirement for qualifying as pure blood. maybe it was all 8 great grandparents being wizards, or all 16 great great grandparents, or all 32 great great great grandparents, or some other rule, but some rule was made.
So a bunch of wizard families that had a high enough proportion of Wizard ancestry declared that they were pure-bloods, even though they may have known about distant Muggle ancestors beyond the limits of their definition of pure-bloods. And the self proclaimed pure-blood families intermarried between themselves (except for a few "black sheep" in every family) for centuries to come.
So in wizard society people talk about pure-blood families that met or exceeded the minimum standards for being defined as pure-blood centuries ago, and have intermarried only with other pure-bloods since then. Thus there are some families that are pure-blood according to the usual wizard definition.
But if some wizard wanted to annoy a pure-blood wizard, or was thinking scientifically, he could accurately say that there were no pure-blood wizards, since it is mathematically impossible for any wizard not to have some Muggle ancestors who lived centuries or millennia earlier, and it is also pretty much impossible for any wizard to know about every single one of their ancestors thirty generations back - well, maybe there are spells for that, but that would a vast amount of information to record and to search for Muggle ancestors.